VIEWS: 8 PAGES: 24 CATEGORY: Education POSTED ON: 5/14/2010 Public Domain
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