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2

Lecture 5: overview

precuationary savings with CARA utility

properties of consumption functions

aggregate implications of idiosyncratic risk (borrowing constraints and
precautionary savings)
—
steady state: Huggett/Aiyagari
(“aggregate precautionary savings”)
—
BC dynamics: Krusell and Smith (next class)
Precuationary Savings: T > 2
first order condition
u c t   β 1  r Ev t 1 1  r z  c   y 

if v t1  0 then there are precautionary
but do we know anything about v  ?

fortunately, if u   0 then v  0 [Sibley (1975)]

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example: u c    exp  γc 
γ
savings
1
r


z
guess and verify that v z    A exp   γ
γ
 1 r 

consumption function
c z  
r
1 r
 1 *
z  r y 


where,
y* 
1 r
  γr 
log E exp 
y
 γr
1  r 
a kind of certainty equivalent
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during
period
t

CARA is an unnatractive assumption... (depends on the question)
...but useful benchmark → helps understand other cases
...can be thought as a local approximation sometimes
...good for aggregation (linearity)

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
aggregation example from problem set (discuss after seeing Aiyagari’s
model)
Infinte vs. Finite periods
infinite:
convenient; sometimes good approximation to finite but long horizon;
dynastic interpretation
bad: long run implications may be very different

finite:
more realistic for many issues
less tractable
4 Analysis of Infinite Period Income Fluctuation Problem
4.1 β (1 + r) = 1

look at example with CARA: upward drift in a

martingale convergence theorem:
Euler implies u ct  must converge

if r = 0 then ct  y  Ey
Shechtman (1975) and Bewley(1976)

if r > 0 then c t  

in both cases at  
saving is very attractive
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4.2 β (1 + r) < 1

analytical properties
— monotonicity of c (z) and a (z)
— borrowing constraint is binding iff z  z *
- think of certainty case
→ region is large ( z * large)
→ it is approached monotonically
- with uncertainty region may be small
and no tendency towards it
— assets are bounded if lim c  0 u c   0 (thus, true for CRRA)
u c 
— if uc   HARA class => c (z) is concave

Bellman equation:
v z   max u c   βEv 1  r z  c   y~ 
c
v is increasing, concave and differentiable.

f.o.c.
uc   β1  r Ev 1  r a  y~
c  a   z, a   0 ; f.o.c. holds with equality if c < z.
=> concavity of v
=> concavity of a   1  r Ev1  r a  ~
y
=> standard consumption problem with two normal goods
=> c (z) and a z  are increasing in z

define
 
u  z *   1  r Ev  ~
y
z  0 ; for z  z c  z
*

*
assets are bounded above
idea: as assets become large uncertainty from income is not important
if absolute risk aversion goes to zero
proof:
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We need to show that there exist a z * such that z max  1  r a z  
y max  z for z  z * . Write the Euler condition as:
Eu c z 
u c z   β 1  r 
u c z max 
u c z max 
Eu c z 
 1 as then we are done.
if we can show that z  
u c z max 
Note that
u c z min  u c z max   z max  z min 
Eu c z 
1


u c z max  u c z max 
u c z max 
We now take lim on both sides. As z → ∞ we can show that a z  and c (z)
go to infinity, therefore. Fortunately we can show that
u c  A
1
u c 
for A > 0. To see this, note that this is equivalent to prooving
u c 
1
u c  A
or
u c  A
1
u c 
We have 1 ≤
u c  A 
Au c  t 
 1  0
dt
u c 
u c 
Au c  t   u c  t 
 1  0
dt
u c  u c  t 
Au c  t 
 1  0
γc  t dt
u c 
A
 1  0 γc  t dt
u c  t 
 1 for all t > 0. The result follows
u c 
since lim c  γc  t   0 for all t  0, A .
the last inequality holds since

computational (Zeldes)
— marginal propensity to consume
— concavity of c (z)
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5

Simulations
Deaton:
borrowing constraint binds infrequently
consumption is smoother at high frequencies
at lower frequencies not so smooth (cannot smooth permanent shocks)
average asset holdings may be small
6 Aiyagari and Huggett


undertakes computational GE exercise with
—
infinite lives
—
CRRA preferences
—
borrowing constraints
—
without capital — Huggett
with capital — Aiyagari which we follow more closely
y t  wlt and l t is random; w is economy-wide wage
6.1 Borrowing constraints

natural borrowing constraint:
c t  0 and no-ponzi implies a t  

wl min
r
Use at   where
 wl min 
, b
 r

  min 

transformed problem: z t
ât  at  
zˆ t  y t  1  r aˆ t  r

convenient: constraints on problem are as before
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6.2 SS equilibrium

laws of motion for distribution
z   1  r az   y 
define the inverse:
   1 z r   1 y r 


z  at
1

 z
yi 
Ft 1 z    pi Ft  a 1 

 




1

r
1

r



a steady state has:

 z
yi 
F z    pi F  a 1 

 




1

r
1

r




N is given by N   l i p i

define steady state equilibrium: three equations in the three unknowns:
K, r and w:
 Az, r , w dF z; r , w   φ  K
r  Fk K , N   
w  FN  K , N 

graph:
1. solve last two equations for w as a function of r: w (r)
2. use this in the .rst equation
 Az, r , wr d z; r , wr   K
to get a relationship between K and r — a steady state supply curve of
capital, Ear  . From the second equation
r  Fk K , N   
we have a demand relationship between K and r: K (r) .
3. find the equilibrium as an intersection of both curves
K r   Ear 
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
properties and comparative statics
A (r) is continuous but may be non-monotonic (effect of w (r))
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A (r) → ∞ as r   1

increase in b: only obvious at r = 0; in practice increases A
increase in uncertainty: obvious? no. but in practice increases A

table II from paper:

wealth distribution: not as skewed

transition? monotonic?
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