Chapter 16 Self Insurance by mzq79210

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									Chapter 16

Self Insurance




                 1
                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.
•   Why do we care about this household
    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
       Inada restriction on preferences.

                                                          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
  – Ad hoc 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
  value of your income)?
• 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:
    with ad hoc borrowing constraint
•     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

                                                          14
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
                                           15
                     (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




                                              18
  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.

                                                                  19
                             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



                                                    20
    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!
Proof by contradiction suppose instead that
  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).
• Instead consider instead utility functions that
  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.


                                             24

								
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