Download Public debt, Finite Horizons in NGM

Public debt, Finite Horizons in NGM
Ec1430 ● Robert Barro ● Harvard University ● 2014
Budget Constraints
 Government bonds, B, infinitesimal maturity, pay real
interest rate r. Government b.c. is
rB  G  T  dB dt .
T is lump-sum taxes. Household assets per person (when
internal loans cancel) are a = k + b . Household b.c. is
da dt  w  r  n  a  c   ,
 is per capita taxes.
 To simplify algebra, assume r constant and n = 0 .
2
Budget Constraints / F.O.C.
 “Lifetime” budget constraint (when TVC holds for
households) is



0
0
0
a0   wt   e rt dt   ct   e rt dt    t   e rt dt ,
a0  k 0  b0 .
 F.O.C. same as before (if c, G separable effects on U):
1 c  dc dt   1    r    .
Different if positive marginal tax rate on saving—for
example, tax on asset income or time-varying consumption
tax. Also effects on labor-leisure choice if taxes on labor or
3
consumption. Assume lump-sum taxes here.
Dynamics
 Dynamic equation for
k̂ is

dkˆ dt  f kˆ  cˆ  gˆ   x  n     kˆ .
Since F.O.C. for dc/dt same as before and path of ĝ is
given, only way that shift between T and dB/dt can affect
equilibrium is through household income effect. From
lifetime b.c., this has to come through term

b0    t   e rt dt ,
or, in aggregate,
0

  B0   t   e dt .
0
rt
4
Ricardian Equivalence
 Can show from government b.c., for any B(0) and path of
T(t), that

    Gt   e dt  lim B  e
0
rt
t 
rt
.
Path of G given. Therefore, no income effect from shift in
B(0) or path of T(t) if chain-letter debt issue (Ponzi game by
government) ruled out. Hence, shift in B(0) or path of T(t)
(for given G path) leaves unchanged equilibrium path of
cˆ, kˆ, r, etc.
 Ricardian equivalence holds in NGM. (Equivalently, tax cut
today is fully saved by households, so that national saving
5
unchanged.)
Alternative approaches
 To study finite-horizon effects, use Blanchard model
(Ch. 3). Use for public debt and also for real money
balances/inflation.
 Alternative approach is overlapping-generations (OLG)
structure, from Samuelson (1958) and Diamond (1965).
See appendix to Ch. 3. In 2-period version, individuals
work, receive wages, consume, and save in period 1;
receive asset income and consume in period 2. May also
pay taxes in period 1 and receive social security in period 2.
Economy goes on forever (no end of world). Individuals
have finite horizons (finite lifetimes) if no concern for future
generations. Barro (1974) adds altruistic linkages across
6
generations—may restore infinite horizon.
Blanchard – Weil
 OLG useful for studying life-cycle, retirement, finite-horizon
effects. But cumbersome for comparative statics.
Blanchard (1985) captures finite-horizon effect in more
tractable way. See extension in Weil (1989) and Ch. 3.
 Blanchard extends NGM to have p per unit of time of dying.
r  f ' kˆ   , as before. Individuals hold annuities—pay r + p

if survive, 0 if die. Attractive asset because people have no
desire to leave anything to descendants. For debts, r + p is
payment on loan secured by life insurance. Annuity-life
insurance companies hold claims on capital—which pay r—
and break even.
7
Blanchard – Weil
 Individual j maximizes expected utility:

EU j   uc j  e  ρt  e  pt dt .
0
e  pt is probability of being alive at t if alive at 0. p constant—
no dependence of mortality rate on age. (Can interpret
“dying” as end of connected dynasty, rather than literal
death.) No n because no concern with children. Blanchard
has n = 0 anyway, but Weil adds n > 0—call this BlanchardWeil model. Assume usual iso-elastic form for uc j  .
8
Blanchard – Weil
 Budget constraint for individual survivor is
da j dt  w j  r  p   a j  c j .
FOC for survivor is the usual:
1 c j  dc j
dt   1    r  p     p 
 1    r    .
 New effects from aggregation of cj over persons of different
ages. Assume people start life with zero assets (no
bequests). Survivors have more assets as they age.
9
Blanchard – Weil
 Assume wj = w, independent of age. Could introduce life
cycle of wages (zero for children, then rising, then falling in
old age). Assume, to simplify, θ=1 (log utility), but could be
dropped. Key to tractable aggregation is independence of
p from age.
 Recall in Ramsey model, with θ=1, consumption is given by
c    n  wealth    n  a  P.V.w
In Blanchard-Weil, n does not appear (because no concern
with children), but  becomes  + p (overall discount rate on
future utils).
10
Blanchard – Weil
 Aggregation of consumption function across ages (along
with assumption that assets = 0 at age 0) leads to key
equation:
1 c  dc dt   r     p  n    p  a c .
Note: c is consumption per capita in overall economy—NOT
consumption of representative person (who vary by age).
 Note in equation that we assumed θ=1. p +n is gross flow
of new persons (n=0 in Blanchard). With n>0, could have
p=0 (infinite lives but new people born). However,
remember that n refers to “disconnected” new persons—
unloved children and immigrants.  +p is MPC out of
assets (independent of r path because θ =1).
11
Blanchard – Weil
 Model still has
dkˆ dt   f kˆ cˆ  x  n     kˆ .
Second equation is different:
1 cˆ dcˆ dt   f ' kˆ   δ  ρ  x   p  n   ρ  p   kˆ cˆ .
 Phase diagram changes. Locus for
dcˆ dt  0 now a
curve that asymptotes to previous vertical line. See figure.
12
Blanchard – Weil
 Phase Diagram in Blanchard Model:
ĉ

f kˆ
cˆ  0
ĉ *

ˆ
k 0
k̂ 0  k̂ *
k̂ gold
k̂
13
Blanchard – Weil
 Steady-state k̂ * lower than before:
 


f ' kˆ *  δ  ρ  x   p  n    ρ  p   kˆ * cˆ * .
Increase in p or n lowers k̂ *. (Note: n refers to population
growth for disconnected people.) Finite-horizon effect is
small—e.g. δ=0.05, ρ=0.02, x=0.02, p=0.02, n=0.01


means that RHS is 0.09  0.0012  kˆ* cˆ* .
 One idea from OLG was equilibrium might have
oversaving— kˆ*  kˆgold . Blanchard shows that finite
horizon is NOT what might produce this result.
14
Blanchard – Weil
 Life-cycle pattern—wages falling by age might do it.
However, typical life cycle has w rising with age for some
time, especially childhood.
 Add public debt to Blanchard model. New persons have no
government bonds but share in future taxes. In key
dynamic equation, can think heuristically of k̂ being
replaced by kˆ  bˆ :
1 cˆ dcˆ dt   f ' kˆ   δ  ρ  x   p  n   ρ  p   kˆ  bˆ cˆ .
(Not fully accurate. b̂ should appear net of expected p.v. of
taxes paid by each living person. This p.v. of taxes
depends also on path of future budget deficits.)
15
Blanchard – Weil
 Can use Blanchard phase diagram readily if bˆ cˆ constant.
Rise in bˆ cˆ shifts locus for dcˆ dt  0 upward. Hence, k̂ *
falls. With finite horizons, people currently alive feel richer
when more government bonds outstanding. Therefore,
want to consume more. Equilibrium requires higher r* and,
hence, lower k̂ * .
 As with finite-horizon effect, impact on r* small. Depends
on  p  n    p   bˆ cˆ  0.0012  bˆ cˆ . (More generally,
 
 
public-debt effects in Blanchard model depend on current
stock of bonds and on expected future budget deficits.)
16