Part IV The savings problem and Bewley mod- els Chapter 16 Self-Insurance 16.1. Introduction This chapter describes a version of what is sometimes called a savings problem (e.g., Chamberlain and Wilson, 2000). A consumer wants to maximize the expected discounted sum of a concave function of one-period consumption rates, as in chapter 8. However, the consumer is cut off from all insurance markets and almost all asset markets. The consumer can only purchase nonnegative amounts of a single risk-free asset. The absence of insurance opportunities induces the consumer to adjust his asset holdings to acquire “self-insurance.” This model is interesting to us partly as a benchmark to compare with the complete markets model of chapter 8 and some of the recursive contracts models of chapter 19, where information and enforcement problems restrict allocations relative to chapter 8, but nevertheless permit more insurance than is allowed in this chapter. A generalization of the single-agent model of this chapter will also be an important component of the incomplete markets models of chapter 17. Finally, the chapter provides our first brush with the powerful supermartingale convergence theorem. To highlight the effects of uncertainty and borrowing constraints, we shall study versions of the savings problem under alternative assumptions about the stringency of the borrowing constraint and alternative assumptions about whether the household’s endowment stream is known or uncertain. – 540 – The consumer’s environment 541 16.2. The consumer’s environment An agent orders consumption streams according to E 0 ∞  t=0 β t u (c t ) , (16.2.1) where β ∈ (0, 1), and u(c) is a strictly increasing, strictly concave, twice con- tinuously differentiable function of the consumption of a single good c.The agent is endowed with an infinite random sequence {y t } ∞ t=0 of the good. Each period, the endowment takes one of a finite number of values, indexed by s ∈ S. In particular, the set of possible endowments is y 1 < y 2 < ··· < y S .Elements of the sequence of endowments are independently and identically distributed with Prob(y = y s )=Π s , Π s ≥ 0, and  s∈S Π s = 1. There are no insurance markets. The agent can hold nonnegative amounts of a single risk-free asset that has a net rate of return r where (1+r)β =1. Leta t ≥ 0 be the agent’s assets at the beginning of period t including the current realization of the income process. (Later we shall use an alternative and common notation by defining b t = −a t +y t as the debt of the consumer at the beginning of period t, excluding the time t endowment.) We assume that a 0 = y 0 is drawn from the time invariant endowment distribution {Π s }. (This is equivalent to assuming that b 0 =0 in the alternative notation.) The agent faces the sequence of budget constraints a t+1 =(1+r)(a t − c t )+y t+1 , (16.2.2) where 0 ≤ c t ≤ a t ,witha 0 given. That c t ≤ a t is the constraint that holdings of the asset at the end of the period (which evidently equal a t+1 −y t+1 1+r )mustbe nonnegative. The constraint c t ≥ 0 is either imposed or comes from an Inada condition lim c↓0 u  (c)=+∞. The Bellman equation for an agent with a>0is V (a)=max c  u(c)+ S  s=1 β Π s V  (1 + r)(a −c)+y s   (16.2.3) subject to 0 ≤ c ≤ a, where y s is the income realization in state s ∈ S. The value function V (a) inherits the basic properties of u(c); that is, V (y) is increasing, strictly concave, and differentiable. 542 Self-Insurance “Self-insurance” occurs when the agent uses savings to insure himself against income fluctuations. On the one hand, in response to low income realizations, an agent can draw down his savings and avoid temporary large drops in con- sumption. On the other hand, the agent can partly save high income realizations in anticipation of poor outcomes in the future. We are interested in the long- run properties of an optimal “self-insurance” scheme. Will the agent’s future consumption settle down around some level ¯c? 1 Or will the agent eventually be- come impoverished? 2 Following the analysis of Chamberlain and Wilson (2000) and Sotomayor (1984), we will show that neither of these outcomes occurs: consumption will diverge to infinity! Before analyzing it under uncertainty, we’ll briefly consider the savings problem under a certain endowment sequence. With a non-random endowment that does not grow perpetually, consumption does converge. 16.3. Nonstochastic endowment Without uncertainty the question of insurance is moot. However, it is instruc- tive to study the optimal consumption decisions of an agent with an uneven income stream who faces a borrowing constraint. We break our analysis of the nonstochastic case into two parts, depending on the stringency of the borrow- ing constraint. We begin with the least stringent possible borrowing constraint, namely, the natural borrowing constraint on one-period Arrow securities, which are risk-free in the current context. After that, we’ll arbitraily tighten the bor- rowing constraint to arrive at the no-borrowing condition a t+1 ≥ y t+1 imposed in the statement of the problem in the previous section. For convenience, we temporarily use our alternative notation. We let b t be the amount of one-period debt that the consumer owes at time t; b t is related to a t by a t = −b t + y t , 1 As will occur in the model of social insurance without commitment, to be analyzed in chapter 19. 2 As in the case of social insurance with asymmetric information, to be ana- lyzed in chapter 19. Nonstochastic endowment 543 with b 0 =0. Here −b t is the consumer’s asset position before the realization of his time t endowment. In this notation, the time t budget constraint (16.2.2) becomes c t + b t ≤ βb t+1 + y t (16.3.1) where in terms of b t+1 , we would express a no-borrowing constraint (a t+1 ≥ y t+1 )as b t+1 ≤ 0. (16.3.2) The no-borrowing constraint (16.3.2) is evidently more stringent than the natural borrowing constraint on one-period Arrow securities that we imposed in chapter 8. Under an Inada condition on u(c)atc = 0, or alternatively when c t ≥ 0 is imposed, the natural borrowing constraint in this non-stochastic case is found by solving (16.3.1) forward with c t ≡ 0: b t ≤ ∞  j=0 β j y t+j ≡ b t . (16.3.3) The right side is the maximal amount that it is feasible to pay repay at time t when c t ≥ 0. Solve (16.3.1) forward and impose the initial condition b 0 =0 toget ∞  t=0 β t c t ≤ ∞  t=0 β t y t . (16.3.4) When c t ≥ 0, under the natural borrowing constraints, this is the only restric- tion that the budget constraints (16.3.1) impose on the {c t } sequence. The first-order conditions for maximizing (16.2.1) subject to (16.3.2) are u  (c t ) ≥ u  (c t+1 ) , =ifb t+1 < b t+1 . (16.3.5) It is possible to satisfy these first-order conditions by setting c t = c for all t ≥ 0, where c is the constant consumption level chosen to satisfy (16.3.4) at equality: c 1 − β = ∞  t=0 β t y t+j . (16.3.6) Under this policy, b t is given by b t = β −t t−1  j=0 β j (c − y j )=β −t   c 1 − β − β t c 1 − β − t−1  j=0 β j y j   = ∞  j=0 β j y t+j − c 1 − β 544 Self-Insurance where the last equality invokes (16.3.6). This expression for b t is evidently less than or equal to b t for all t ≥ 0. Thus, under the natural borrowing constraints, we have constant consumption for t ≥ 0, i.e., perfect consumption smoothing over time. The natural debt limits allow b t to be positive, provided that it is not too large. Next we shall study the more severe ad hoc debt limit that requires −b t ≥ 0, so that the consumer can lend, but not borrow. This restriction will inhibit consumption smoothing for households whose incomes are growing, and who therefore are naturally borrowers. 3 16.3.1. An ad hoc borrowing constraint: non-negative assets We continue to assume a known endowment sequence but now impose a no- borrowing constraint (1 + r) −1 b t+1 ≤ 0 ∀t ≥ 0. To facilitate the transition to our subsequent analysis of the problem under uncertainty, we work in terms of a definition of assets that include this period’s income, a t = −b t +y t . 4 Let (c ∗ t ,a ∗ t ) denote an optimal path. First order necessary conditions for an optimum are u  (c ∗ t ) ≥ u   c ∗ t+1  , =ifc ∗ t <a ∗ t (16.3.7) for t ≥ 0. Along an optimal path, it must be true that either (a) c ∗ t−1 = c ∗ t ;or (b) c ∗ t−1 <c ∗ t and c ∗ t−1 = a ∗ t−1 , and hence a ∗ t = y t . Condition (b) states that the no-borrowing constraint binds only when the con- sumer desires to shift consumption from the future to the present. He will desire to do that only when his endowment is growing. According to conditions (a) and (b), c t−1 can never exceed c t . The reason is that a declining consumption sequence can be improved by cutting a marginal unit of consumption at time t −1 with a utility loss of u  (c t−1 ) and increasing 3 See exercise 16.1 for how income growth and shrinkage impinge on con- sumption in the presence of an ad hoc borrowing constraint. 4 When {y t } is an i.i.d. process, working with a t rather than b t makes it possible to formulate the consumer’s Bellman equation in terms of the single state variable a t , rather than the pair b t ,y t . We’ll exploit this idea again in chapter 17. Nonstochastic endowment 545 consumption at time t by the saving plus interest with a discounted utility gain of β(1+r)u  (c t )=u  (c t ) >u  (c t−1 ), where the inequality follows from the strict concavity of u(c)andc t−1 >c t . A symmetrical argument rules out c t−1 <c t as long as the nonnegativity constraint on savings is not binding; that is, an agent would choose to cut his savings to make c t−1 equal to c t as in condition (a). Therefore, consumption increases from one period to another as in condition (b) only for a constrained agent with zero savings, a ∗ t−1 −c ∗ t−1 = 0. It follows that next period’s assets are then equal to next period’s income, a ∗ t = y t . Solving the budget constraint (16.2.2) at equality forward for a t and rear- ranging gives ∞  j=0 β j c t+j = a t + β ∞  j=1 β j y t+j . (16.3.8) At dates t ≥ 1forwhicha t = y t , so that the no-borrowing constraint was binding at time t − 1, (16.3.8) becomes ∞  j=0 β j c t+j = ∞  j=0 β j y t+j . (16.3.9) Equations (16.3.8) and (16.3.9) contain important information about the opti- mal solution. Equation (16.3.8) holds for all dates t ≥ 1atwhichtheconsumer arrives with positive net assets a t − y t > 0. Equation (16.3.9) holds for those dates t at which net assets or savings a t −y t are zero, i.e., when the no-borrowing constraint was binding at t − 1. If the no-borrowing constraint is binding only finitely often, then after the last date t−1 at which it was binding, (16.3.9) and the Euler equation (16.3.7) imply that consumption will thereafter be constant at a rate ˜c that satisfies ˜c 1−β =  ∞ j=0 β j y t+j . In more detail, suppose that an agent arrives in period t with zero savings and that he knows that the borrowing constraint will never bind again. He would then find it optimal to choose the highest sustainable constant consumption. This is given by the annuity value of the tail of the income process starting from period t, x t ≡ r 1+r ∞  j=t (1 + r) t−j y j . (16.3.10) In the optimization problem under certainty, the impact of the borrowing con- straint will not vanish until the date at which the annuity value of the tail (or 546 Self-Insurance remainder) of the income process is maximized. We state this in the following proposition. Proposition 1: Given a borrowing constraint and a nonstochastic endow- ment stream, the limit of the nondecreasing optimal consumption path is ¯c ≡ lim t→∞ c ∗ t =sup t x t ≡ ¯x. (16.3.11) Proof: We will first show that ¯c ≤ ¯x. Suppose to the contrary that ¯c>¯x. Then conditions (a) and (b) imply that there is a t such that a ∗ t = y t and c ∗ j >x t for all j ≥ t. Therefore, there is a τ sufficiently large that 0 < τ  j=t (1 + r) t−j  c ∗ j − y j  =(1+r) t−τ  c ∗ τ − a ∗ τ  , where the equality uses a ∗ t = y t and successive iterations on budget constraint (16.2.2). The implication that c ∗ τ >a ∗ τ constitutes a contradiction because it violates the constraint that savings are nonnegative in optimization problem (16.2.3). To show that ¯c ≥ ¯x, suppose to the contrary that ¯c<¯x. Then there is an x t such that c ∗ j <x t for all j ≥ t, and hence ∞  j=t (1 + r) t−j c ∗ j < ∞  j=t (1 + r) t−j x t = ∞  j=t (1 + r) t−j y j ≤ a ∗ t + ∞  j=t+1 (1 + r) t−j y j , where the last weak inequality uses a ∗ t ≥ y t . Therefore, there is an >0and ˆτ>tsuch that for all τ>ˆτ , τ  j=t (1 + r) t−j c ∗ j <a ∗ t + τ  j=t+1 (1 + r) t−j y j − , and after invoking budget constraint (16.2.2) repeatedly, (1 + r) t−τ c ∗ τ < (1 + r) t−τ a ∗ τ − , Nonstochastic endowment 547 or, equivalently, c ∗ τ <a ∗ τ − (1 + r) τ −t . We can then construct an alternative feasible consumption sequence {c  j } such that c  j = c ∗ j for j =ˆτ and c  j = c ∗ j +  for j =ˆτ . The fact that this alternative sequence yields higher utility establishes the contradiction. More generally, we know that at each date t ≥ 1 for which the no-borrowing constraint is binding at date t−1, consumption will increase to satisfy (16.3.9). The time series of consumption will thus be a discrete time ‘step function’ whose jump dates t coincide with the dates at which x t attains new highs: t = {t : x t >x s ,s<t}. If there is a finite last date t, optimal consumption is a monotone bounded sequence that converges to a finite limit. In summary, we have shown that under certainty the optimal consumption sequence converges to a finite limit as long as the discounted value of future income is bounded. Surprisingly enough, that result is overturned when there is uncertainty. But first, consider a simple example of a nonstochastic endowment process. 16.3.2. Example: Periodic endowment process Suppose that the endowment oscillates between one good in even periods and zero goods in odd periods. The annuity value of this endowment process is equal to x t    t even = r 1+r ∞  j=0 (1 + r) −2j =(1−β) ∞  j=0 β 2j = 1 1+β , (16.3.12a) x t    t odd = 1 1+r x t    t even = β 1+β . (16.3.12b) According to Proposition 1, the limit of the optimal consumption path is then ¯c =(1+β) −1 . That is, as soon as the agent reaches the first even period in life, he sets consumption equal to ¯c forevermore. The associated beginning-of-period assets a t fluctuates between (1 + β) −1 and 1. The exercises at the end of this chapter contain more examples. 548 Self-Insurance 16.4. Quadratic preferences It is useful briefly to consider the linear-quadratic permanent income model as a benchmark for the results to come. Assume as before that β(1 + r)=1 and that the household’s budget constraint at t is (16.3.1). Rather than the no-borrowing constraint (16.3.2), we impose that 5 E 0  lim t→∞ β t b 2 t  =0. (16.4.1) This constrains the asymptotic rate at which debt can grow. Subject to this constraint, solving (16.3.1) forward yields b t = ∞  j=0 β j (y t+j − c t+j ) . (16.4.2) We alter the preference specification above to make u(c t ) a quadratic func- tion −.5(c t −γ) 2 ,whereγ>0 is a ‘bliss’ consumption level. Marginal utility is linear in consumption: u  (c)=γ −c. We put no bounds on c;inparticular,we allow consumption to be negative. We allow {y t } to be an arbitrary stationary stochastic process. The weakness of constraint (16.4.1) allows the houshold’s first-order con- dition to prevail with equality at all t ≥ 0: u  (c t )=E t u  (c t+1 ). The linearity of marginal utility in turn implies E t c t+1 = c t , (16.4.3) which states that c t is a martingale. Combining (16.4.3) with (16.4.2) and tak- ing expectations conditional on time t information gives b t = E t  ∞ j=0 β j y t+j − 1 1−β c t or c t = r 1+r   −b t + E t ∞  j=0  1 1+r  j y t+j   . (16.4.4) 5 The natural borrowing limit assumes that consumption is nonnegative, while the model with quadratic preferences permits consumption to be nega- tive. When consumption can be negative, there seems to be no natural lower bound to the amount of debt that could be repaid, since more payments can always be wrung out of the consumer. Thus, with quadratic preferences we have to rethink the sense of a borrowing constraint. The alternative (16.4.1) allows negative consumption but limits the rate at which debt is allowed to grow in a way designed to rule out a Ponzi-scheme that would have the consumer always consume bliss consumption by accumulating debt without limit. [...]... of how the 554 Self-Insurance marginal utility of consumption will otherwise turn negative at large consumption levels Thus, our understanding of the remarkable result in Proposition 2 is aided by considering the inevitable ratchet effect upon consumption implied by the first-order condition for the agent’s optimal intertemporal choice 16. 8 Concluding remarks This chapter has maintained the assumption... infinity (We return to this result in chapter 17 on incomplete market models.) Though assets diverge to infinity, they do not increase monotonically Since assets are used for self-insurance, we would expect that low income realizations are associated with reductions in assets To show this point, suppose to the contrary that even the lowest income realization y1 is associated with nondecreasing assets; that... + ys , = (16. 5.3) s=1 where the last equality is first-order condition (16. 5.2 ) when the nonnegativity constraint on savings is not binding and after using β −1 = (1 + r) Since V [(1 + r)(a − c) + ys ] ≤ V [(1 + r)(a − c) + y1 ] for all s ∈ S, expression (16. 5.3 ) implies that the derivatives of V evaluated at different asset values are equal to each other, an implication that is contradicted by the... to rule out negative consumption 16. 5 Stochastic endowment process: i.i.d case With uncertain endowments, the first-order condition for the optimization problem (16. 2.3 ) is S β(1 + r)Πs V (1 + r)(a − c) + ys , u (c) ≥ (16. 5.1) s=1 with equality if the nonnegativity constraint on savings is not binding The Benveniste-Scheinkman formula implies u (c) = V (a), so the first-order condition can also be written... and constant risk-free one-period net interest Exercises 557 rate of r that satisfies (1 + r)β = 1 The consumer’s budget constraint at t is at+1 = (1 + r)(at − ct ) + yt+1 , subject to the initial condition a0 = y0 One-period assets carried (at − ct ) over into period t + 1 from t must be nonnegative, so that the no-borrowing constraint is at ≥ ct At time t = 0 , after y0 is realized, the consumer... finite limit, which is ruled out 16. 6 Stochastic endowment process: general case The result that consumption diverges to infinity with an i.i.d endowment process is extended by Chamberlain and Wilson (2000) to an arbitrary stationary stochastic endowment process that is sufficiently stochastic Let It denote the information set at time t Then the general version of first-order condition (16. 5.4 ) becomes u (ct... time t Then the general version of first-order condition (16. 5.4 ) becomes u (ct ) ≥ E u (ct+1 ) It , (16. 6.1) where E(·|It ) is the expectation operator conditioned upon information set It Assuming a bounded utility function, Chamberlain and Wilson prove the following result, where xt is defined in (16. 3.10 ): Proposition 2: If there is an > 0 such that for any α ∈ P α ≤ xt ≤ α + It < 1 − for all It... consumption, but the former is also associated with a larger rise in marginal utility as compared to the drop in marginal utility of the latter To set today’s marginal utility of consumption equal to next period’s expected marginal utility of consumption, the consumer must therefore balance future states with expected declines in consumption against appropriately higher expected increases in consumption for other... i.i.d case 549 Equation (16. 4.4 ) is a version of the permanent income hypothesis and tells the consumer to set his current consumption equal to the annuity value of his nonj ∞ 1 human (−bt ) and ‘human’ (Et j=0 1+r yt+j ) wealth We can substitute this consumption rule into (16. 3.1 ) and rearrange to get ∞ bt+1 = bt + rEt j=0 1 1+r j yt+j − (1 + r) yt (16. 4.5) Equations (16. 4.4 ), (16. 4.5 ) imply that... infinity After invoking the Benveniste-Scheinkman formula, first-order condition (16. 5.1 ) can be rewritten as S u (c) ≥ S β (1 + r) Πs u (cs ) = s=1 Πs u (cs ) , (16. 5.4) s=1 where cs is next period’s consumption if the income shock is ys , and the last equality uses (1 + r) = β −1 It is important to recognize that the individual will never find it optimal to choose a time-invariant consumption level for . Part IV The savings problem and Bewley mod- els Chapter 16 Self-Insurance 16. 1. Introduction This chapter describes a version of what is sometimes called a savings problem (e.g., Chamberlain and. is allowed in this chapter. A generalization of the single-agent model of this chapter will also be an important component of the incomplete markets models of chapter 17. Finally, the chapter. constraint (16. 2.2) becomes c t + b t ≤ βb t+1 + y t (16. 3.1) where in terms of b t+1 , we would express a no-borrowing constraint (a t+1 ≥ y t+1 )as b t+1 ≤ 0. (16. 3.2) The no-borrowing constraint (16. 3.2)