Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 16 pps

A consumer wants to maximize theexpected discounted sum of a concave function of one-period consumption rates, as in chapter 8.. To highlight the effects of uncertainty and borrowing cons

The savings problem and Bewley els

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 theexpected 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 andalmost all asset markets The consumer can only purchase nonnegative amounts

of a single risk-free asset The absence of insurance opportunities induces theconsumer to adjust his asset holdings to acquire “self-insurance.”

This model is interesting to us partly as a benchmark to compare with thecomplete markets model of chapter 8 and some of the recursive contracts models

of chapter 19, where information and enforcement problems restrict allocationsrelative to chapter 8, but nevertheless permit more insurance than is allowed inthis 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 supermartingaleconvergence theorem

To highlight the effects of uncertainty and borrowing constraints, we shallstudy versions of the savings problem under alternative assumptions aboutthe stringency of the borrowing constraint and alternative assumptions aboutwhether the household’s endowment stream is known or uncertain

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16.2 The consumer’s environment

An agent orders consumption streams according to

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 {yt} ∞

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 y1 < y2< · · · < yS Elements

of the sequence of endowments are independently and identically distributed

with Prob(y = y s) = Πs, Πs ≥ 0, and s ∈SΠs = 1 There are no insurancemarkets

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 Let a 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=−at +y t

as the debt of the consumer at the beginning of period t , excluding the time

t endowment.) We assume that a0 = y0 is drawn from the time invariantendowment distribution {Πs} (This is equivalent to assuming that b0 = 0 inthe alternative notation.) The agent faces the sequence of budget constraints

a t+1 = (1 + r) (a t − ct ) + y t+1 , (16.2.2)

where 0≤ ct ≤ at , with a0 given That c t ≤ at is the constraint that holdings

of the asset at the end of the period (which evidently equal a t+1 1+r −y t+1) must be

nonnegative The constraint c t ≥ 0 is either imposed or comes from an Inada

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

“Self-insurance” occurs when the agent uses savings to insure himself againstincome fluctuations On the one hand, in response to low income realizations,

an agent can draw down his savings and avoid temporary large drops in sumption On the other hand, the agent can partly save high income realizations

con-in anticipation of poor outcomes con-in the future We are con-interested con-in the run properties of an optimal “self-insurance” scheme Will the agent’s futureconsumption 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!

long-Before analyzing it under uncertainty, we’ll briefly consider the savingsproblem 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 tive to study the optimal consumption decisions of an agent with an unevenincome stream who faces a borrowing constraint We break our analysis of thenonstochastic 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, whichare risk-free in the current context After that, we’ll arbitraily tighten the bor-

instruc-rowing constraint to arrive at the no-borinstruc-rowing condition a t+1 ≥ yt+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

ana-with b0= 0 Here −bt 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

ct + b t ≤ βbt+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

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) at c = 0 , or alternatively when

ct ≥ 0 is imposed, the natural borrowing constraint in this non-stochastic case

is found by solving ( 16.3.1 ) forward with c t ≡ 0:

Solve ( 16.3.1 ) forward and impose the initial condition b0= 0 to get

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 {ct} sequence The

first-order conditions for maximizing ( 16.2.1 ) subject to ( 16.3.2 ) are

u  (c t)≥ u  (c

t+1 ) , = if b 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:

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

−bt ≥ 0, so that the consumer can lend, but not borrow This restriction will

inhibit consumption smoothing for households whose incomes are growing, andwho 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=−bt +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



, = if c ∗ 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 sumer desires to shift consumption from the future to the present He will desire

con-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 at rather than b t makes itpossible to formulate the consumer’s Bellman equation in terms of the single

state variable a t , rather than the pair b t, yt We’ll exploit this idea again inchapter 17

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) and c t −1 > ct A symmetrical argument rules out c t −1 < ct 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 ranging gives

At dates t ≥ 1 for which at = y t, so that the no-borrowing constraint was

binding at time t − 1, (16.3.8) becomes

opti-arrives with positive net assets a t − yt > 0 Equation ( 16.3.9 ) holds for those

dates t at which net assets or savings a t−ytare 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 1−β˜c =∞

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

con-remainder) of the income process is maximized We state this in the followingproposition.

Proposition 1: Given a borrowing constraint and a nonstochastic ment stream, the limit of the nondecreasing optimal consumption path is

endow-¯

c ≡ lim

t →∞ ct = supt

xt ≡ ¯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 > xt for all j ≥ t Therefore, there is a τ sufficiently large that

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 itviolates 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

xt such that c ∗ j < xt for all j ≥ t, and hence

or, equivalently,

c τ < a ∗ τ − (1 + r) τ −t 

We can then construct an alternative feasible consumption sequence {cj} such

that c j = c ∗ j for j = ˆτ and cj = c ∗ j +  for j = ˆ τ 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 : xt > xs, 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 consumptionsequence converges to a finite limit as long as the discounted value of futureincome is bounded Surprisingly enough, that result is overturned when there isuncertainty But first, consider a simple example of a nonstochastic endowmentprocess

16.3.2 Example: Periodic endowment process

Suppose that the endowment oscillates between one good in even periods andzero goods in odd periods The annuity value of this endowment process is equalto

t even= β

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

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 that5

E0

lim

This constrains the asymptotic rate at which debt can grow Subject to this

constraint, solving ( 16.3.1 ) forward yields

func-allow consumption to be negative We func-allow {yt} to be an arbitrary stationary

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 away designed to rule out a Ponzi-scheme that would have the consumer alwaysconsume bliss consumption by accumulating debt without limit

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 human (−bt ) and ‘human’ (E t

non-∞

j=0

1

1+r

j

y t+j) wealth We can substitute

this consumption rule into ( 16.3.1 ) and rearrange to get

equiv-The next section shows that these outcomes will change dramatically when

we alter the specification of the utility function to rule out negative consumption

16.5 Stochastic endowment process: i.i.d case

With uncertain endowments, the first-order condition for the optimization



(1 + r)(a − c) + ys, (16.5.1)

with equality if the nonnegativity constraint on savings is not binding The

Benveniste-Scheinkman formula implies u  (c) = V  (a) , so the first-order

condi-tion can also be written as

where a  s is next period’s assets if the income shock is y s Since β −1 = (1 + r) ,

V  (a) is a nonnegative supermartingale By a theorem of Doob (1953, p 324),7

V  (a) must then converge almost surely The limiting value of V  (a) must be zero based on the following argument: Suppose to the contrary that V  (a) converges to a strictly positive limit That supposition implies that a converges

to a finite positive value But this implication is immediately contradicted by the

nature of the optimal policy function, which makes c a function of a , together with the budget constraint ( 16.2.2 ): randomness of y s contradicts a finite limit

for a Instead, V  (a) must converge to zero, implying that assets diverge to

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 con- trary that even the lowest income realization y1 is associated with nondecreasing

assets; that is, (1 + r)(a − c) + y1≥ a Then we have

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 strict concavity of V

The fact that assets converge to infinity means that the individual’s sumption also converges to infinity After invoking the Benveniste-Scheinkman

con-formula, first-order condition ( 16.5.1 ) can be rewritten as

where c  s is next period’s consumption if the income shock is y s, and the last

equality uses (1 + r) = β −1 It is important to recognize that the individualwill never find it optimal to choose a time-invariant consumption level for the

7 See the appendix of this chapter for a statement of the theorem

... realizations are associated with reductions in assets To show this point, suppose to the con- trary that even the lowest income realization y1 is associated with nondecreasing

assets;... individual’s sumption also converges to infinity After invoking the Benveniste-Scheinkman

con-formula, first-order condition ( 16. 5.1 ) can be rewritten as

where c  s... return to this result in chapter 17 on incomplete market models.)Though assets diverge to infinity, they not increase monotonically Since

assets are used for self-insurance, we would expect