# Chapter 16 Self Insurance by mzq79210

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```									Chapter 16

Self Insurance

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Objective
• Describe the solution to a household
savings problem
– An agent wants to maximize the present value
of consumption
– However, agent is cut off from all insurance
markets and almost all asset markets.
– One savings technology: Risk-asset
– Household can only accumulate positive
amounts of this asset.
2
Objective (continued)
•   Absence of insurance means that
household must self-insure against
fluctuations in his income.
problem?
1. Benchmark for comparison with the
complete markets model.
2. Stepping stone to describing general
equilibrium in Bewley type incomplete
markets economies.
3
The economy
• Preferences
∞
E0 ∑ β t u ( ct ) ,                 (16.2.1)
t =0

• Budget constraint
at +1 = (1 + r )( at − ct ) + yt +1.       (16.2.2 )
or
at +1 − yt +1
+ ct = at
1+ r
4
Distribution of endowments
• The endowment yt is a finite S state
markov chain. We will assume for the time
being that the sequence of endowments
are i.i.d. with
Pr ob( y = ys ) = Π s
Πs ≥ 0

∑Π
s∈S
s   =1
5
Rate of return on risk free asset
• Assumption on magnitude of rate of return
of risk free asset:

(1 + r ) β = 1
• This turns out to make sense in many
complete market economies.
• In this chapter and chapter 17 we will find
that this relationship has some interesting
properties!
6
Restrictions on consumption
•    Consumption satisfies:

0 ≤ ct ≤ at
•    Why?
1. Holdings at the end of the period of the
asset must be non-negative (at +1 − yt +1)/(1+ r) ≥ 0
2. Non-negative consumption follows from an

7
Bellman equation for household

⎧           S
⎫
V ( a ) = max ⎨u ( c ) + ∑ β ∏ s V ⎡(1 + r )( a − c ) + ys ⎤ ⎬ ,
⎣                       ⎦       (16.2.3)
c
⎩          s =1                                ⎭
subject to 0 ≤ c ≤ a,

8
Properties of the Value function
• If the utility function is increasing, strictly
concave and differentiable, the value
function will have the same properties.

9
Properties of the solution
•    Self insurance
1. When endowment is high agent saves some of his
endowment.
2. When endowment is low agent runs down his
savings.
–    Several outcomes are possible depending on
the probability structure:
1. Consumption may settle down at a positive level: c
2. Consumption may go to zero.
3. Consumption my diverge to infinity!
10
Non-stochastic Endowment
• Two types of borrowing constraints
– Natural borrowing constraint
• The problem (16.2.3) imposes an ad hoc
borrowing constraint.

11
Non-stochastic endowment:
with a natural borrowing constraint
• What happens with a natural borrowing
constraint (can borrow up to the present
• Perfect consumption smoothing.

∞
c
= ∑ β yt
t

1 − β t =0

12
Alternative specification of budget
constraint
• Let
at = −bt + yt
• the borrowing constraint is now:

−1
(1 + r ) bt +1 ≤ 0
Note that bt denotes borrowing!

ct + bt ≤ β bt +1 + yt      13
Non-stochastic endowment:
•     FONC for household problem:
u '(ct* ) ≥ u '(ct*+1 ), = if ct* < at*
•     With a sure but fluctuating endowment
stream the optimal path must satisfy one
of the two conditions below:

(a )      ct*−1 = ct*

(b)       c*
t −1   < c and c
*
t
*
t −1   = a , and hence a = yt
*
t −1
*
t

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Implications of these two conditions
1. A declining consumption pattern is not an
optimal solution.
2. Similarly if the non-negativity constraint
is not binding an increasing consumption
pattern is not an optimal solution.
3. For an increasing consumption pattern to
be optimal the agent must have:
(i ) at*−1 = ct*−1
and
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(ii ) a = y
*
t
*
t
Implications of the two conditions
• Suppose an agent arrives in period t with
zero savings and he knows that the
borrowing constraint will never bind again.
Then the optimal policy is to find the
highest sustainable constant consumption
level given by:
∞
r
∑ (1 + r ) y j .
t− j
xt ≡
1 + r j =t
16
Under certainty the optimal
consumption sequence
converges to a finite limit when
the discounted value of future
endowments is bounded

17
• Proposition: Given a borrowing constraint
and a nonstochastic endowment stream,
the limit of the nondecreasing optimal
consumption path is:

c ≡ lim c = sup xt ≡ x .
t →∞
*
t                      (16.3.11)
t

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Stochastic endowment process
• With uncertain endowments the FONC for
the households problem is:
S
u ' ( c ) ≥ ∑ β (1 + r ) ∏ s V ' ⎡(1 + r )( a − c ) + y s ⎤,
⎣                        ⎦    (13.5)
s =1

With equality if the borrowing constraint is
not binding.

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FONC
• Using the Benviniste Sheinkman formula:

u '( c ) = V '( a )
We have:

S
V ' ( a ) ≥ ∑ β (1 + r ) ∏ s V ' ( a 's ),   (16.5.2 )
s =1

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Assets converge to infinity!
• From Doob (1953 p. 324) we know that
V’(a) converges almost surely.
• In fact, the limiting value must be zero!
V’(a) converges to a strictly positive limit.
This then implies that a converges to a
positive finite value.
But, this is contradicted by the budget
constraint (16.2.2).
21
General properties of the
solution when β (1 + r ) = 1
1. Assets don’t converge to infinity
monotonically. When income is low
agents will draw down their assets.
2. Consumption tends to infinity.
3. Household will never find it optimal to
choose a time-invariant Consumption
level for all future periods.

22
Intuition for the consumption result
• Under quadratic utility don’t get the result.
(Certainty equivalence).
satisfy: u’’’>0.
• Jensen’s inequality implies:
∑   s
Π s u '(cs ) >u '   (∑ Π c )
s   s s

• Then (16.5.1) plus Benveniste-Scheinkman:
S
c < ∑ Π s cs'
s =1
23
Intuition
• Previous equation says that consumption
today is strictly lower than expected
consumption tomorrow.
• This ratchet effect explains why
consumption tends to ever higher levels.
• Moreover, consumption can not converge
to a finite limit.

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