Slides 10

Debraj Ray

Columbia, Fall 2013

Behavioral Poverty Traps

Some Literature

Three models of behavioral traps

0-0

Alternative Approaches to the Study of Poverty

Constraints

absence of credit

absence of insurance

nonconvexities (nutrition, health, education)

Psychology

failed aspirations

informational biases

temptation, lack of self-control

0-1

Two Examples from Developing Countries

Poor forego profitable small investments

Agricultural investment in Ghana (Udry-Anagol, 2006)

Fertilizer use in Kenya (Duflo-Kremer-Robinson, 2010)

Microenterprises in Sri Lanka (Mel-McKenzie-Woodruff, 2008)

Public distribution debate

Public food distribution system in India

Huge debate on food versus cash transfers

Impulsive spending from cash (Khera 2011 survey)

0-2

Self-Control or Just Present Bias?

Demand for commitment products in LDCs.

Lockboxes in the Gambia (Shipton, 1992)

Commitment savings in the Philippines (Ashraf-Karlan-Yin, 2006)

ROSCAS (Aliber, 2001, Gugerty, 2001, 2007, Anderson-Baland, 2002)

Pressures to share

Extended-family demands on wealth

(Platteau 2000, Hoff-Sen 2006, Brune et al 2011)

0-3

Self-Control

Intuitive idea:

Ability to follow through on an intended plan

(operationally, match a choice made with full precommitment)

External versus internal devices.

External: locked savings, retirement plans, Roscas etc.

Internal: the use of psychological private rules (Ainslee).

see Strotz (1956), Phelps-Pollak (1968), or Laibson (1997).

0-4

Other possibilities:

Dual self (Thaler-Shefrin 1981, Fudenberg-Levine 2006)

Resisting temptation (Gul-Pesendorfer, 2003)

Ainslee private rules as self-discovery (Ali 2011)

Literature on the particular question pursued here:

Direct assumptions on preferences

Banerjee-Mullainathan, 2010

Capital market imperfections that generate non-homotheticity

Bernheim-Ray-Yeltekin, 1999, 2013

0-5

Preferences: The “Shape” of Temptation

Based on Banerjee-Mullainathan (2010) [BM]

“The link to poverty within this framework comes from as-suming that the fraction of the marginal dollar that is spent ontemptation goods can depend on the level of consumption.”

Divide assets A into consumption c and bequest b:

A = c+ b

while new assets A′ are a random variable given by

A′ = f (b, θ).

View as wealth plus labor income, so f (0, θ) generally positive.

BM writec = x+ z

where x is a standard good and z is a temptation good.

0-6

Two-period model. In period 2, agent maximizes

U(x2) + V (z2), subject to x2 + z2 = A′.

Let x(A′) and z(A′) be resulting consumption functions.

In period 1, agent does not value V (z2), so maximizes

U(x1) + V (z1) + δEU(x(A′)) = U(x1) + V (z1) + δEU(x(f (b, θ)),

subject to the constraint x1 + z1 + b = A.

Leads to the first-order condition:

U ′(x1) = V ′(z1) = δEU ′(x(f (b, θ))f ′(b, θ)x′(f (b, θ))= δEU ′(x(f (b, θ))f ′(b, θ)[1− z′(f (b, θ))],

which BM call the modified Euler equation.

Obviously, similar equation would hold in multi-period model.

0-7

Some Immediate Implications

The desire to commit.

Doesn’t fully emerge in this framework because commitmentblocks both x- and z-consumption.

But would commit if it could protect x-consumption and hinderz-consumption.

Example: purchase of durable goods today.

The effect of sin taxes.

Imagine a future tax on period-2 consumption of z.

Effect on savings will depend on what happens to derivativex′(A′).

Could go up or down.

0-8

The Shape Assumption

Note that the formulation does not restrict curvature of x(A′)or z(A′) in any way.

Main assumption. z is strictly concave.

Temptation matters less at the margin as assets go up.

BM justify this by saying that:

temptations are visceral and kick in more at low incomes.

temptation goods are more divisible

“it may be easier for a rich person to say no to a relative whowants a few hundred dollars . . . than for a poor person to refuseone who wants just a couple of dollars for a meal.”

Without commenting on any of this, let’s just say it’s an em-pirical question.

0-9

Implications of the Shape Assumption

(Under additional presumption that Euler equation still valid.)

The poor appear more impatient.

Proof: effective discount factor given by

δ = δx′(A′).

Possible anti-smoothing of consumption.

Raise future labor income by uniformly raising f (0, θ).

In standard model with no z, consumption today ↑ (smoothing).

Here this might flip: countervailing effect given by the fact thatz′(A′) flattens.

So modified Euler equation could generate more saving.

0-10

Poverty traps.

Reconsider maximization problem:

U(x1) + V (z1) + δEU(x(f (b, θ)),

subject to the constraint x1 + z1 + b = A.

Write c1 = x1 + z1 let W be indirect utility function, so maximize

W (c1) + δEU(x(f (b, θ)),

subject to c1 + b = A.

Recall monotonicity lemma: b(A) nondecreasing in A.

If x is concave, then the problem is concave and no jump in b.

But if z is concave, b could jump up.

Interpreted as a “poverty trap.”

0-11

Possible Lack of Prudence.

An increase in income uncertainty encourages savings in a safetechnology:

Needs a third-derivative restriction on the utility function.

No longer sufficient in this case.

Investment Scale.

Suppose an investment is feasible, has a given return and upperbound on scale.

In standard model, the upper bound is unimportant in decisionto invest.

Here an increase in the bound can matter, as it lowers the“temptation derivative” z′(A′).

0-12

Adverse effects of credit.

Today’s self can become worse off if tomorrow’s self has accessto credit.

Need a three-period model (at least) for this.

BM show that with declining temptations, period-0 self mightallow period-1 self to have a big loan (and so get temptation todecline) rather than a small loan which will all be blown on thez-good.

0-13

All the results depend on assuming that the poor are moretempted than the rich.

This begs the main question.

Bernheim, Ray and Yeltekin (1999, 2013) take a different ap-proach.

They assume that the underlying model is homothetic in pref-erences.

The only non-homothetic feature is an imperfect credit market.

0-14

Assets and Incomes

Accumulation

At = ct +At+1

α.

Imperfect credit market

At ≥ B > 0.

Interpretation: A = financial assets + pv of labor income

P =α

α− 1y,

andB = Ψ(P )

e.g.,B = ψP for some ψ ∈ (0, 1]

0-15

Preferences u(c) = c1−σ/(1− σ), for σ > 0.

u(c0) + β∞∑t=1

δtu(ct), 0 < β < 1.

Standard model: β = 1.

If δα > 1 [growth] and µ ≡ 1α(δα)1/σ < 1 [discounting], then

At+1 = (δα)1/σAt

ct = (1− µ)At.

−→ Ramsey policy.

If β < 1, optimal plan is time-inconsistent.

0-16

Policies and Values

Policy φ specifies continuation asset At+1 after every history

Atct

!_At+1

A0 A1

History

CurrentFuture

. . .

And so generates values and payoffs:

V (ht) ≡ u(ct) + δu(ct+1) + δ2u(ct+2) + . . .

P (ht) ≡ u(ct) + β[δu(ct+1) + δ2u(ct+2) + . . .

]= u(ct) + βδV (ht.φ(ht))

No self-starvation: c ≥ νA for some ν tiny but positive.

0-17

Equilibrium

Following the policy is better than trying something else.

P (ht) ≥ u(A(ht)− x

α

)+ βδV (ht.x) for every x ∈ [B,αA(ht)].

B AA1 A2

Equilibrium Values

0-18

Self-Control Definition

Self-control at A:

⇒ Accumulation at A in some equilibrium.

Strong self-control at A:

⇒ At →∞ from A, in some equilibrium.

No self-control at A:

⇒ No accumulation at A in any equilibrium.

Poverty trap at A:

⇒ Slide to credit limit B from A in every equilibrium.

0-19

Self-Control and No Self-Control

A

A'

B

B No self control

Self controlNo self control

0-20

Uniformity and Nonuniformity

Uniform case:

Self control at every A, or its absence at every A.

Nonuniform case:

Self-control at A, no self-control at A′.

0-21

Uniformity and Nonuniformity

A

A'

B

B

0-22

Uniformity and Nonuniformity

A

A'

B

B

0-23

Uniformity and Nonuniformity

A

A'

B

B

0-24

Uniformity and Nonuniformity

Uniform case:

Self control at every A, or its absence at every A.

Nonuniform case:

Self-control at A, no self-control at A′.

Theorem. Suppose no credit constraints, so that B = 0.

Then every case is uniform.

Poverty bias not built in; contrast Banerjee and Mullainathan(2010).

0-25

Credit Constraints and Non-Uniformity

B > 0 destroys scale-neutrality (in A), but how exactly?

Some intuition:

Think of the consequences of a lapse in self-control.

More severe when the individual has more assets; hence moreto lose.

Problem:

Not bad as intuition, but unfortunately does not work.

Severity isn’t monotone in assets.

To see this, first we understand the structure of worst punish-ments.

0-26

The Structure of Lowest Values

V(A)

B

H(A)

L(A)

AA' A*

Theorem. If A′ > B is continuation for A∗ under lowest valueat A∗, then A′ is followed by value H−(A′).

0-27

The Structure of Lowest Values

V(A)

B

H(A)

L(A)

AA' A*A"

Theorem. If A′ > B is continuation for A∗ under lowest valueat A∗, then A′ is followed by value H−(A′).

u(c′′t )+βδBlue = u(c′t)+βδOrange⇒ u(c′′t )+ δBlue < u(c′t)+ δOrange.

0-28

Lowest Values

Simple structure. Following a deviation:

One more binge, then the highest-value program.

Like Abreu penal codes, but for entirely different reasons.

Argument also reveals why L(A) jumps up occasionally.

0-29

maximize u(A− x/α) + βδL(x), say max at x = A.

V(A)

B

H(A)

L(A)

AA'A

Not possible; get a contradiction:

u(ct)+βδBlue ≤ u(c′t)+βδOrange⇒ u(ct)+ δBlue < u(c′t)+ δOrange.

0-30

maximize u(A− x/α) + βδL(x), say max at x = A.

V(A)

B

H(A)

L(A)

AA' A A

So A > A′, and u(ct) + βδBlue = u(c′t) + βδOrange.

By concavity of u, A′ may need to jump up, so L(A) jumps too.

0-31

Argument So Far

The problem of internal self-control is both simple and complex.

Simple: what happens after lapse of control is easy to describe.

Lapse followed by one round of high c, then back to best path.

Complex: jump in worst values makes comparative statics hard.

As wealth goes up, can get cycles of control / failure of control.

0-32

Markov Equilibrium: Values and Continuations

A AB B SS

V A'

0-33

0.5 0.6 0.7 0.8 0.9 1 1.1 1.20.5

0.6

0.7

0.8

0.9

1

1.1

A

A’

Markov Perfect Equilibria: Savings Function, !=0.75, "=1.28, #=0.8

Markov45o line

0-34

Illustration of the nonuniform case:

A AB B

A'A' Markov continuation asset Maximal continuation X(A)

But the simulations suggest that this is not true with history-dependent strategies.

0-35

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

Savings, !=0.75

45o line

Highest saving

Best SPE

Ramsey

Markov

Worst SPE

0-36

3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4 3.45 3.53

3.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

3.5

Savings, !=0.75

45o line

Highest saving

Best SPE

Ramsey

Markov

0-37

1 2 3 4 5 6 7 83

4

5

6

7

8

9

10

11

12

13

14Equilibrium Values, !=0.75

SPE best

SPE worst

Ramsey

Markov

0-38

Theorem. [Central Result]. In the non-uniform case,

There is A1 > B such that every A ∈ [B,A1) exhibits a povertytrap.

There is A2 ≥ A1 such that every A ≥ A2 exhibits strong self-control.

A

A'

B A1

X(A)

A2

0-39

Proof Outline I. The Poverty Trap

X(A): maximum wealth choice. Then X(A) < A close to B.

AB SM µA1

X(A)

A1450

0-40

Proof Outline II. Strong Self-Control, contd.

A** A***A

[(!1)kA**, (!2)kA***][(!1)k+1A**, (!2)k+1A***]

(!1)m(!2)nA

X(A) !1 = A**/B, !2 = A***/B

0-41

Some Implications

1. Easier Access to Credit Has Ambiguous Effects

Conventional theory: more abundant credit reduces saving.

Implications here are more nuanced.

Modified neutrality: only B/A matters.

Easier credit (lower B) reduces A1 and A2 thresholds:

More individuals successfully exercise self-control

Offsetting effect: those who fall into trap will fall further.

Summary: ambiguous effects, depending on where you start.

0-42

2. The Demand for Commitment Devices

Demand for external commitment devices by poor households.

Surprisingly little evidence that this demand is more widespread.

(But: Ariely-Wertenbroch 2002, Beshears-Choi-Laibson-Madrian 2011)

Need some reliance on internal mechanisms (value of flexibility).

But external devices undermine efficacy of internal mechanisms.

Who demands external devices?

The asset-poor, and the income-rich if B ∝ permanent income.

The asset-rich or the income-poor prefer internal mechanisms.

Income-rich generally also asset-rich, so net effect is ambiguous.

0-43

3. Designing Accounts to Promote Saving

Example: retirement savings programs.

Significant variation in lock-up across plans

In addition, large variation in stringency of contributions.

Recall: lock-up has both upside and downside.

Programs that capitalize on upside while avoiding downside?

Idea: lock up funds until some target, then remove the lock.

Can (should) allow each individual to select personal target.

0-44

To formalize, use taste shock for uncertain environment:

u(c, η) = ηc1−σ

1− σ,

Lock-up account that unlocks once a threshold is reached.

Threshold slightly higher than the threshold that permits accu-mulation.

If lower, the agent will slide back once the account is unlocked.

Note: nowhere close to solving the optimal design problem.

0-45

Alternative A. commitment savings up to threshold, full release thereafter.

Current Assets

Val

ues

RamseyBest Value without LockboxBest Value with Lockbox

B AT Current AssetsV

alue

s

RamseyBest Value without LockboxBest Value with Lockbox

AT

Both the lock-up and the release are important . . .

0-46

Alternative B: commitment savings forever, principal always locked.

Alternative C: usual saving after threshold, commitment principal locked.

Current Assets

Val

ues

Best Value without LockboxBest Value, Principal Accessible after ThresholdBest Value, Principal not AccessibleBest Value, Principal Locked after Threshold

ATB Current Assets

Val

ues

Best Value without LockboxBest Value, Principal Accessible after ThresholdBest Value, Principal not AccessibleBest Value, Principal Locked after Threshold

AT

Note: Alternative C can be worse than Alternative B.

0-47

4. Asset-Specific MPCs

Hatsopoulos-Krugman-Poterba (1989), Thaler (1990), Laibson (1997)

A = financial assets + permanent income.

Jump in financial assets

B/A = B/(financial assets + permanent income).

B/A falls: can switch from decumulation to accumulation.

So low MPC from financial assets.

Jump in income. If B/(perm inc) constant, B/A ↑.

High MPC in non-uniform case.

At best B unchanged; then identical MPCs.

0-48

Summary

We know that a failure of self-control can lead to poverty.

Is the opposite implication true?

Model constructed for scale-neutrality:

Result isn’t “built-in” by presuming that the poor are temptedmore.

Ainslee’s personal rules as history-dependent equilibria

Structure of optimal personal rules is remarkably simple:

Deviations entail “falling off” the wagon, then “climbing backon”.

0-49

Main result: ability to impose self-control rises with wealth.

In fact, the model generates a poverty/self-control trap.

Novel policy implications:

Among them: interplay between external and internal commit-ments

External self-control devices can undermine internal self-control

Lock-box savings accounts with self-established targets and un-locking of principal may be particularly effective devices for increas-ing saving

0-50

Poverty/temptation link one of three behavioral poverty traps.

The Information Trap

Information a commodity that poor households cannot afford.

Arsenic contamination of groundwater

Madajewicz et al (2007) document large well-switching afterinfo campaign

HIV/AIDS incidence by age of partner.

Dupas (2011) field experiment on info to teenagers in Kenya.

Also possibility of internal coordination failures with multiplechallenges:

Diarrhea, respiratory disease, malaria, groundwater arsenic

0-51

The Aspirations Trap

Appadurai (2004), Ray (1998, 2006), Genicot-Ray (2013)

A person’s aspirations affects her incentives to invest.

Investments determine growth and income distribution.

But aspirations are not formed in isolation.

The income distribution in turn shapes aspirations.

Two-way process of aspirations formation and growth.

0-52

Society with large number of single-parent single-child families.

Each person has lifetime income or wealth y:

y = c+ k, and z = f (k),

where z is child’s lifetime wealth. Parent payoff given by

u(c) + v(z) + ω(z, a).

u: increasing, smooth, strictly concave with u′(0) =∞.

v: paternalistic altruism, increasing, smooth, strictly concavewith v′(0) =∞.

ω: “aspirations utility,” payoff from child’s wealth relative to a,the aspiration of the parent.

0-53

Write ω(z, a) as

ω(z, a) = s(a)w(z/a),

where w is a purely relative component and s picks up scale effects.

We maintain the following assumptions throughout:

[W.1] w(x) is smooth, with w′(x) > 0.

[W.2] w′′(x) > 0 for x < 1 and w′′(x) > 0 for x > 1.

[W.3] s(a) > 0 for a > 0, and is nondecreasing, smooth andconcave.

These curvature assumptions similar to Koszegi and Rabin (2006).

0-54

z

w(a, z)

a

0-55

The Formation of Aspirations

Two alternative approaches:

A private viewpoint: personal experiences determine futuregoals.

A social viewpoint: the lives of others determine future goals.

That is the approach we take here:

a = Ψ(y,F ),

where F is the society-wide distribution of societal wealth.

[A.1] Ψ is continuous, nondecreasing in y, and Ψ(y,F ) ∈ Range(F ).

0-56

Consider some particular processes of aspirations formation:

Common Aspirations:

Ψ(y,F ) = ψ(F ).

Stratified Aspirations:

Ψ(y,F ) = ai

for individuals with income y in quantile i

where ai summarizes distribution in that quantile.

Upward-Looking Aspirations:

Ψ(y,F ) =

∫∞y xdF (x)

1− F (y).

which is the conditional mean income above y.

0-57

Local Aspirations With Population Neighborhoods.

People draw upon those in some cognitive window around them.

E.g., suppose weight placed only on the surrounding d (income)percentiles:

Ψ(y,F ) =1

d

∫ H(y)

L(y)xdF (x),

where L(y) and H(y) are the edges of cognitive window at y.

Local Aspirations With Income Neighborhoods.

Weight only on incomes within an interval N(y):

Ψ(y,F ) =1

F (N(y))

∫N(y)

xdF (x),

where F (N(y)) has the obvious meaning.

0-58

Equilibrium

Suppose that the distribution of current wealth is given by Ft.

Then each person in t with wealth yt has at = Ψ(yt,Ft).

A policy φ maps y and a to z for the next generation.

Equilibrium from F0 is a sequence {Ft} and policy φ such that

(i) For every t and y, a = Ψ(y,Ft), and z = φ(y, a) maximizes

u(y− f−1(z)) + v (z) + s(a)w (z/a) over z ∈ [0, f (y)].

(ii) Ft+1 generated from Ft and φ; that is, for each z ≥ 0,

Ft+1(z) = Probt{y|φ(y, Ψ(y,Ft)) ≤ z},

0-59

Aspirations, Wealth and Growth: Partial Equilibrium

FOC: v′ (z) +s(a)

aw′ (z/a) = u′(y− f−1(z))f−1

′(z)

za

v(z) + ω(z, a)

z1z2

u(y) - u(y-f -1(z))

At most one solution z that exceeds a. See point z1.

There could be a number of them below a; see point z2.

0-60

Pick solution that yields the highest payoff.

Generically unique.

Say that aspirations are satisfied if there is a solution z ≥ a.

Aspirations are frustrated if there is z < a.

(We will see that once aspirations are frustrated at a, everyoptimal solution for every a′ > a will fall short of a′.

When are aspirations satisfied, and when are they frustrated?

0-61

Changes in Aspirations

Change aspirations exogenously (e.g., rise of mass media).

Theorem. For given y, there is a unique threshold a∗ belowwhich aspirations are satisfied, and above which they are frustrated.Once frustrated, wealth cannot increase as aspirations continue togrow.

Proof Outline.

For small a, aspirations satisfied and for large a, frustrated.

To show that the threshold is unique:

Lemma. No map that links a to an optimal z can jump upwards.

Idea: utility above aspirations is strictly concave.

0-62

Back to main proof: say frustration at a1.

Let z1 < a1 be optimal. a1 ↑ a2, with z2 optimal at a2. Then

u(c1) + v(z1) + ω(z1, a1) ≥ u(c2) + v(z2) + ω(z2, a1),

and likewise

u(c2) + v(z2) + ω(z2, a2) ≥ u(c1) + v(z1) + ω(z1, a2).

Adding these inequalities and transposing terms,

ω(z1, a1)− ω(z2, a1) ≥ ω(z1, a2)− ω(z2, a2).

For a small increase in aspirations from a1 to a2, Lemma impliesmax{z1, z2} < a1 < a2 for any optimal choice z2 at a2. But over thiszone, the cross partial derivative

∂2ω(z, a)

∂z∂a

is strictly negative. That implies z1 ≥ z2 .

0-63

A graphical illustration

za z1z2

Note: crossover from satisfaction to frustration will usually (butnot always) occur with a discrete fall in investment.

0-64

Monotonicity in the satisfaction zone?

Not necessarily.

Evaluate the sign of the cross-partial ∂2ω/∂z∂a:

∂2ω(z, a)

∂z∂a= φ′(a)w′(z/a)− φ(a)w′′(z/a)z/a2,

where φ(a) = s(a)/a.

This is unambiguously negative in the frustration zone, so dis-courages accumulation.

However, ambiguous sign in the satisfaction zone

So chosen wealth can increase or decline with aspirations.

0-65

Changes in Wealth

Usual monotonicity argument, nothing new here:

Theorem. For given a, there is a unique threshold value ofcurrent wealth below which aspirations are frustrated, and abovewhich they are satisfied. Optimally chosen wealth is nondecreasingin current wealth.

Go further by studying the effect on growth rates of wealth.

Let u(c) = c1−σ/(1− σ) and v(z) = ρz1−σ/(1− σ) for (σ, ρ)� 0.

Suppose that the production function is linear: f (k) = (1+ r)k.

Define g ≡ (z/y)− 1. So individual now chooses g to maximize

1

1− σ

(y

[r− g1+ r

])1−σ+

ρ

1− σ([1+ g]y)

1−σ + s(a)w([1+ g]y/a).

0-66

Recallling φ(a) = s(a)/a as the “normalized” scale effect, FOC:

(r− g)−σ(1+ r)σ−1 − ρ(1+ g)−σ = φ(a)w′ ([1+ g]y/a) yσ.

g

lhs, rhs

rg1 g2

rhs

-1

lhs

g

lhs, rhs

rg1g2

rhs

-1

lhs

Frustrated aspirations Satisfied aspirations

0-67

Inspection of the figures suggests the following:

Theorem. If aspirations are frustrated at y, the growth rate ofincome declines as incomes fall below y.

frustration from “failed aspirations”.

What of initial wealth levels for which aspirations are satisfied?

Answer depends on RHS of the FOC.

Panel B of figure drawn on the presumption that:

[W.4] w′(z/a)zσ is declining in z when z > a.

Theorem. Under [W.4], growth rates decline as incomes in-crease, once aspirations are met.

Complacency from “satisfied aspirations”.

0-68