Fiscal Policy
(204A -3e)-P.1
JMany issues in practice – not all suitable for representative agent modeling
J*D8C01;48BBD4B
- Effects of government spending – real spending on good & services
- Optimal fiscal policy with regard to government spending
- B4=27<0A:=4DCA0;8CHA4BD;CB>=58=0=28=6)820A380=neutrality/equivalence theorem
- Effects of tax distortions – incentive effects
JMore elaborate models needed for
-)438BCA81DC8>=0;C0G0C8>= – requires model with heterogenous population
- *C018;8I0C8>=?>;82H– requires model with stochastic disturbances
- Government debt and social security – better with generational turnover
J'DC;8=4
I. Effects of government spending assuming lump-sum taxes; with note on optimal policy
II. )820A380=neutrality
III. Examples with tax distortions – examples only.
(204A -3e)-P.2
Fiscal Policy I: Government Spending
JBBD<?C8>=(D1;82B?4=38=6 ?4A4558284=2HD=8C>5;01>A
- Assume spending is tax-58=0=243+ ;D<?-sum.
- Here abstract from productivity and population growth (could be added)
- Best interpret as government consumption. [Public capital would require different model.]
J)4B>DA242>=BCA08=C?4A-20?8C0 k = f ( k ) − δk − c − G
- Phase diagram with c(k)-;8=4
c ( k ) = f ( k ) − δk − G .
J(A454A4=24Butility over private consumption U = ∫0 e
∞
− ρt
Higher G shifts the c(k)-line down.
u (c)dt .
- +A40C public spending as exogenous. *C0=30A3<02A> view. Endogenous in public finance.)
!0<8;C>=80=
H (c,k,λ ,t) = e−ρt u(c) + λ ⋅[ f (k) − δk − c − G]
-&>C4 4=C4AB>=;H8=C>C742>=BCA08=C*0<4D;4A4@D0C8>=B0BF8C7>DC6>E4A=<4=C
c = 1 ( f ' (k ) − δ − ρ )
c θ (C )
-*C403HBC0C42>=38C8>=B
k = 0
c = 0
c* = f (k * ) − δk * − G*
f ' (k * ) = δ + ρ
-"<?;843?70B43806A0<B8G43:*-;8=4*785CB8=2:-line whenever G(t) changes.
(204A -3e)-P.3
Application #1: Permanent Increase
J8=3"=BC0=C9D<?C>C74=4FBC403HBC0C4 >E4A=<4=CB?4=38=670B=>8<?02C>=20?8C0;BC>2:=>8<?02C>=
C748=C4A4BCA0C42A>F3B>DC?A8E0C42>=BD<?C8>=>=4-for-one.
(204A -3e)-P.4
Application #2: Temporary Increase
J A0?7B*44)><4A86=4GCB;834
J#4H8340>;;>F38554A4=CL0AA>FBM?70B43806A0<B0BC8<4?0BB4B
- Until t02>=><H0CC748=8C80;BC403HBC0C4F8C7=>A<0;
- Interval [t0C1/%>E4<4=CB8=C74?70B43806A0<5>;;>F2:5D=2C8>=F8C778674A - After t1%>E4<4=CB8=C74?70B4 diagram follow c(k) function with normal G.
J*>;E4102:F0A3B
-)4CDA=C>C74BC403HBC0C48=C74;>=6AD=%DBC14>=C74N=>A<0;OB033;4?0C70CC 1.
- Determine c(0) such that movements during [t0C1] end on the saddle path at t1.
J8=3+4<?orary increase government spending has
-0C4<?>A0AH=460C8E445542C>=20?8C0;022D<D;0C8>=2A>F38=6>DC
- a temporary negative impact on consumption.
- a temporary positive impact on interest rates.
Optional Exercise*D??>B40CC8<4C0C746>E4A=ment announces that spending will increase
?4A<0=4=C;HBC0AC8=60CC8<4C1. Determine the effects.
(204A -3e)-P.5
)><4A86
(204A -3e)-P.6
Note on Preferences and Optimal Policy
J*D??>B48=38E83D0;B70E4?A454A4=24B>E4A?A8E0C40=3?D1;822>=BD<?C8>=
∞
U = ∫ e−ρt u(c,G)dt
0
-*C0=30A30??A>0278=?D1;8258=0=24*>280;?;0==4A>?C8<8I4B>E4A
-!0<8;C>=80=
H (c,G,k,λ ,t) = e− ρt u(c,G) + λ ⋅[ f (k) − δk − c − G]
-??;HC74%0G8<D<(A8=28?;4F8C7CF>27>824E0A801;4B
∂H = e − ρt ∂u − λ = 0 and ∂H = e − ρt ∂u − λ = 0
∂c
∂c
∂G
∂G
∂
u
∂
u
'?C8<0;8CHA4@D8A4B /
= 1D=8C<0A68=0;A0C4>5BD1BC8CDC8>=
∂c ∂G
J)4BD;C'?C8<0;C>38E834C>total consumption c˜ = c + G into private and public components
such that each provides the same utility on the margin.
JG?A4BBC74<>34;8=C4A<B>5C>C0;2>=BD<?C8>=
- write c = c( c˜ ) G = G( c˜ ) 0=3 u˜ ( c˜ ) = u[c( c˜ ),G(c˜ )]
∞
-4;50A4?A>1;4<<0G8<8I4U = ∫ e
0
− ρt
u˜ (c˜ )dt s.t. k = f (k ) − δk − c~
J"=C4A?A4C0C8>=+74>?C8<0;6A>FC7<>34;F8C7>DC6>E4A=<4=CB42C>A20=148=C4A?A4C43 as
model in which public spending is subsumed into private spending.
(204A -3e)-P.7
Fiscal Policy II: Government debt and deficits
JBBD<?C8>= 8E4=?0C7>5B?4=38=620=1458=0=243F8C7C0G4B+>A341C
- >E4A=<4=C1D364C4@D0C8>=
-"=C46A0C4>E4AC8<4
D(t) = D(0) ⋅ e
- Apply present value factors
∫ 0t r(v)dv
p0,t = e
t
+ ∫ (G(s) − T (s)) ⋅ e
0
− ∫ 0t r(v)dv
t
∫ st r(v)dv
ds
t
p0,t D(t) = D(0) + ∫ p0,sG(s)ds − ∫ p0,sT (s)ds
0
0
- Impose the transversality condition lim t→∞ p0,t D(t) = 0 . Obtain the
J >E4A=<4=COB8=C4AC4<?>A0;1D364C2>=BCA08=C
∞
∞
0
0
D(0) + ∫ p0,sG(s)ds = ∫ p0,sT (s)ds
-"=C4A?A4C0C8>=Initial debt and the present value of spending must be backed by the present value of taxes.
J;08<)820A380==4DCA0;8CH8=0=28=6>5?D1;82B?4=38=63>4B=>C<0CC4A
(204A -3e)-P.8
(A>>5>5)820A380==4DCA0;8CH
J*C4?"=38E83D0;1D364C4@D0C8>=F8C7C0G4B
"=C4AC4<?>A0;1D364C2>=BCA08=C
∞
∞
∞
0
0
0
a(0) + ∫ p0,sw(s)ds = ∫ p0,sc(s)ds + ∫ p0,sT (s)ds
J*C4?><18=4F8C7C746>E4A=<4=C1D364C2>=BCA08=C
∞
∞
∞
0
0
0
a(0) + ∫ p0,sw(s)ds = ∫ p0,sc(s)ds + D(0) + ∫ p0,sG(s)ds
J*C4?"=38E83D0;0BB4CB20?8C0;government bonds-A8C4 #0– D0BBD<8=6$
∞
∞
∞
0
0
0
K(0) + ∫ p0,sw(s)ds = ∫ p0,sc(s)ds + ∫ p0,sG(s)ds
J8=3+0G4B0=36>E4A=<4=C1>=3B20=24;>DC
PV of consumption = Private resources – PV of public spending
J"=C4A?A4C0C8>=
1. )820A380==4DCA0;8CH 8E4=C74B?4=38=6?0C7C0G4B0=3345828CB70E4=>8<?0ct on
individual decisions.
2. )820A380=4@D8E0;4=24+0G-financed and deficit-financed government spending have
equivalent effects on individual decisions.
3. >E4A=<4=C1>=3BB7>D;3&'+142>=B834A43=4CF40;C7 (Barro)
(204A -3e)-P.
J(A02C820;A4;4E0=24Are budget deficits really irrelevant?
1. Good intuition – taxpayers own the government debt.
Modifications needed with C0G38BC>AC8>=B58=8C4;8E4B2A438C<0A:4C8<?4A542C8>=BD=24AC08=CH01>DC
8=2834=24>55DCDA4C0G4B4C2;0C4A – all best understood as departures from the benchmark.
2. Key insight for more elaborate modelsif the income effects of tax policy 20=24;>DCthe substitution
effects remain *hift of emphasisB8=24B to tax distortions / incentive / Lsupply sideM effects.
3. Property of representative 064=C<>34;Baccept or use different model.
Fiscal Policy III: Tax Incentives
JG0<?;4 from )><4A*D??>B4C0G4B0A48<?>B43>= capital income.
-"=38E83D0;B0E8=6B
a= w(t) + (1− τ ) ⋅ r(t) ⋅ a(t) − c(t)
'?C8<0;8CH2>=38C8>=
-*C403HBC0C42>=38C8>=B
c* = f (k * ) − δk * − G* where G* = τ [ f ' (k * ) − δ ]k *
(1− τ )[ f '(k * ) − δ ] = ρ
-(70B43806A0<B:*-line shifts to the left; c(k) as before +0G4BF8C78=24=C8E445542CB3><0CC4A
J&>C4>=3424=CA0;8I0C8>= -74=C0G4B0A438BC>AC8>=0AH<0A:4C4@D8;81A8D<8B=>C(0A4C>-optimal
0==>CDB4 planning problem to compute equilibrium (A3E0=243add Limplementability constraintsM)
(204A -3e)-P.10
Example with labor-leisure choice
JG0<?;4 not8=)><4Asetting with labor-leisure choice and τ(t)labor income tax rate
∞
U = ∫ e − ρt u(c,1− l)dt with unit time allocated to l = labor, (1-l) = leisure
%0G8<8I4
BD1942CC>
-!0<8;C>=80=
0
a= (1− τ (t))⋅ w(t) ⋅l(t) + r(t) ⋅ a(t) − c(t)
H (c,l,a, λ ,t) = e − ρt u(c,1− l) + λ ⋅[(1− τ )wl + ra − c]
-??;HC74%0G8<D<(A8=28?;4F8C7CF>27>824E0A801;4B
∂H = e − ρt ∂u(c,1−l) − λ = 0 and ∂H = e − ρt (− ∂u(c,1−l) ) + (1− τ )wλ = 0
∂c
∂c
∂l
∂l
∂u(c,1−l)
(1−
τ
)w
=
) depends on the tax rates.
Individual optimality2>=38C8>= ∂u(c,1−l)
∂c
∂l
- >E4A=<4=CB?4=3B 70Brevenues T (t) = τ (t) ⋅ w(t) ⋅l(t) D = G + rD − τ ⋅ w ⋅l and
K = a− D = w ⋅l + r ⋅ (a − D) − c − G = F(K,l) − δK − c − G = l ⋅[ f (k) − δk] − c − G
+axes cancel out for20?8C0;022D<D;0C8>=1DC05542C;01>ABD??;H.
1 [ f ' (k) − δ − ρ ]
J*D??>B4u() is separableEuler equation c = θ (C)
c
J*D??>B4 τ) are constant (:2;converge to a steady state with f ' (k * ) − δ = ρ
∂u(c * ,1−l* ) /∂l
∂u(c * ,1−l* ) /∂c
= (1− τ ) ⋅ FL (K * ,l * ) = (1− τ ) ⋅ w(k * ) c* = l * ⋅[ f (k * ) − δk * ] − G .
- *CA82C;H2oncave utility ∂c* / ∂τ < 0 ∂l * / ∂τ < 0
*
*
J>=CA0BCC>F4;50A4proble<<0G,BC K = F(K,l) − δK − c − G ∂u(c* ,1−l* ) /∂l = w(k * )
∂u(c ,1−l ) /∂c
+0G38BC>AC8>=BA43D24F4;50A4 as compared to an allocation with lump-sum taxes.
(204A -3e)-P.11
Learning Objectives for Fiscal Policy
J??;843 Ability to do fiscal policy applications
- Determine the impact of government spending in the optimal growth model.
Problem sets for practice.
J>=24?CD0;
- Know the impact of government spending in the optimal growth model.
- #=>FF74=)820A380==4DCA0;8CH0??;84B.
- Know that distortionary taxes create complications.