Contents

1 Introduction 21.1 Previous Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Approach and Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Key Notation and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Model 62.1 The Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Period 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Period 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Implications of the Modified Euler Equation 83.1 Demand for Durables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Demand for Commitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 ’sin’ taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 The Shape of Temptation 94.1 Declining (DTC) and Non-Declining (NDTC) Temptations . . . . . . . . . . . . 104.2 Relation to Hyperbolic Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2.1 Hyperbolic Discounting with two heterogeneous goods . . . . . . . . . . . 114.2.2 Hyperbolic Discounting with two homogenous goods . . . . . . . . . . . . 11

12

13

14

5 Implications of DTC and NDTC

6 Empirical Tests of the Model and Extensions

7 Conclusion

8 Critique 14

1

Shape of Temptation: Diamond Summary

Diamonds (are forever)

January 26, 2015

1 Introduction

A common perception of the poor is that they are myopic. By myopic, we mean having a strongpreference for consuming today rather than tomorrow under the standard model of consumptiondecision-making. In more technical jargon, we mean that the poor are perceived to have a highdiscount rate. A superficial analysis of their behavior appears to support this conclusion.For instance, Aleem (1990) shows they borrow at high interest rates. Karlan and Mullainathan(2009) give evidence that small-time fruit vendors in India finance their businesses by paying 5%interest a day over nearly a decade. In their investment decisions, it also appears that the poorare opting for present consumption over feasible high return investment activities (see Mel,McKenzkie and Woodruff (2009); Lee, Kremer and Robinson (2009); and Udry and Anagol(2006)). This could only be plausible under the assumption that individuals are consumingat steady state levels, or if the poor are rapidly emerging from poverty. However, in theaforementioned examples the poor remain poor.

There are other potential explanations that involve making non-standard assumptions aboutthe utility function, but they fall short empirically. One is that the poor are prevented fromcutting present consumption because there is a minimum amount they must consume. Empiricalwork shows the consumption of the poor varies a lot (Collins et. al. (2009)) and that they spenda substantial portion of their income on non-necessity goods such as cigarettes, alcohol, andexpensive but not nutritious foods (Banerjee and Duflo (2007)). Alternatively, it could bethat high mortality causes this high discount rate. This explanation is inadequate becausedeath rates at age 30 or 40 are similar among the very poor and wealthier people. Anotherseemingly plausible explanation is that the interest rates are not realized due to high defaultrates, but Banerjee (2004) and Aleem (1990) show that the default rates of lenders to the poorare extremely low.

This paper provides an alternative explanation: that amount of the marginal dollar spenton goods that provide utility in the present, but don’t in calm, calculated foresight (i.e. goodsthat are tempting) decreases with overall consumption (i.e. as one gets richer). This suggeststhat the level of myopia in the poor is lower than perceived.

1.1 Previous Literature

This expands on a literature that models myopic behavior within a broader set of time incon-sistent preferences. This includes Shefrin and Thaler (1981), Laibson (1997), and O’Donoghueand Rabin (1999). Another idea is that people, including the poor, are myopic in some domainsbut farsighted in others (Collins et. al. (2009).

1.2 Approach and Summary of Results

Key Assumption 1: There are two types of goods:

2

1. Non-temptation: Goods that generate utility before they are consumed and while theyare consumed.

2. Temptation: Goods that generate utility only at the point of consumption.

To understand the difference, consider a person that is going to a Duke basketball game.Say this person is asked before the game if they would like to buy a funnel cake at the game. Hemay respond, ”No. I have cut unhealthy food from my diet it will do me no good.” But at thegame, as the warm and sugary aroma of funnel cake ascends into his nostrils, he is possessedby temptation, and proceeds to devour the first one he can get his hands on. As he is eating it,he is enjoying (getting utility) from every bite. In this context, the funnel cake is an exampleof a temptation good. One can imagine similar scenarios involving goods such as alcohol andtobacco. The point is that he derived no utility when considering his future self consuming thefunnel cake. Before the game he did not want his future self to spend money on it. If it was anon-temptation good, his two ”selves” would have agreed on the utility on consuming it.

Key Assumption 2: Declining temptation. The fraction of the marginal dollar that isspent on temptation goods is declining as the level of consumption rises. This is in contrast toa scenario where the fraction either stays the same or goes up with consumption.

Consider two people: Shuxian, who lives on $30 a day; and Daniel, who lives on $2 a day.Assume that a funnel cake costs $1, and that Shuxian and Daniel are equally tempted by it.Giving in to this temptation is more costly for Daniel than it is for Shuxian. The point is thatthe poor will spend a larger portion of their marginal dollar on ”temptation goods.”

In the authors’ own words, ”The main point of this paper is that the assumption of decliningtemptation has a number of striking and novel implications that are relevant to understandingthe savings, credit, risk-taking and investment choices made by the poor and how the poor gettreated in asset markets.”

Extends logic of the standard hyperbolic model:The hyperbolic model focuses on time inconsistency in the level, as opposed to the composi-

tion, of consumption. The time inconsistency stems from different discount factors at differentperiods of time. In this model, consumption is partitioned into spending on temptation goods,and spending on non-temptation goods.

Modified Euler Equation: The New Palgrave Dictionary of Economics defines an EulerEquation as, ”a difference or differential equation that is an intertemporal first-order conditionfor a dynamic choice problem. It describes the evolution of economic variables along an optimalpath.” This paper uses a modified version of the standard Euler Equation that includes a termfor spending on the temptation good. This is significant because there is ”dropped utility”from the part of the expenditure of later period goods that was not valued by the self in earlierperiods.

Key assumption 3: z(c), the function that quantifies the amount of total consumptionthat does to temptation goods, is concave. Equivalently, z’(c) is decreasing.

One implication of this assumption and the structure of the modified Euler equation is thatif we use the traditional Euler equation, the poor will appear to be more myopic than the non-poor, even if they are not. This model suggests that myopic behavior is both a cause and aresult of poverty.

Another implication is that this concave ”temptation tax” alone can generate poverty trapthat those below some strictly positive critical level of initial wealth w will be stuck in. Thosewith wealth below w will either dissave or save very little, while those with wealth above thatlevel will save much more. The intuition is that above w there is a bigger incentive to save,because the ”temptation tax” goes down as wealth goes up. In contrast, below w any incentivesto save are smaller than those of the temptation tax.

3

A concave temptation tax also affects people’ responses to uncertainty. This manifestsitself in two ways. The first is that because they fear their future selves will spend savingson temptation goods, they don’t hold assets to buffer themselves from the shocks they face.The second is that an increase in variance can reduce precautionary savings, regardless ofassumptions of prudence in the utility functions u(x) and v(z).

The framework presented in this paper also has implications for the scale of chosen invest-ments. In standard utility theory, the maximum scale of investments is not considered. It makesit difficult to theoretically explain investments that are capped, like fertilizer, which is cappedby the amount of land owned. These difficulties are mitigated when declining temptations areintroduced.

There are also interesting results regarding the demand for credit. With declining temp-tations, people could prefer to take bigger loans over smaller loans in future periods. Underconstant temptations, over-borrowing can be harmful to the initial period self by exaggeratingthe difference in consumption between the selves in later periods, and by borrowing more thanthe initial self would want. Declining temptations creates an offsetting force: borrowing a largeramount would cause the agent to spend more of the loan on non-temptation goods.

The paper also considers the supply side of credit. With the preferences that are idiosyncraticto this model, monopolistic lenders may have an incentive to deny individuals theoreticallyprofitable loans to maintain power over borrowers.

4

1.3 Key Notation and Terms

ct Total goods consumed, xt + zt, in period t

xt Non-temptation goods consumed in period t

zt Temptation goods consumed in period t

z(c) Temptation goods consumed as a function of total consump-tion

x(c) Non-temptation goods consumed as a function of total con-sumption

U() Utility function for non-temptation goods

V () Utility function for temptation goods

W () Indirect utility function

δ Intertemporal discount rate

hatδ Intertemporal discount rate observed under Standard EulerEquation

R Gross interest rate in Euler Equation

β The hyperbolic coefficient in the Hyperbolic Discount Model

y1 A deterministic labor income in period 1

y2(θ′) Labor income in period 2, is a function of shock term θ′

wt Wealth or amount saved at time t

θ A random shock parameter for the income generating func-tion

f(., .) The income generating function or returns on savings in pe-riod 2

p The sin tax. The price of z goods in terms of x goods wherethe price of x goods has been normalized to 1.

I Amount invested

Lt Amount the agent borrows from lender in period t.

Myopic Having a strong preference for immediate consumption overconsumption in future periods. In the discount models, thisequates to having high β and/or δ parameter values.

Time preference The relative valuation placed on a good at an earlier datecompared with its valuation at a later date.

Temptation goods Goods that generate utility only at point of consumption

Non-temptation goods All other goods services.

Declining temptations (DTC) The fraction of the marginal dollar that is spent on temp-tation goods declines as the level of total consumption(orincome) rises.

Non-declining temptations (NDTC) The fraction of the marginal dollar that is spent on tempta-tion goods is either constant or increasing(modeled in thispaper) as the level of consumption or income rises.

Temptation Tax Income spent tomorrow that is partly spent on temptationgoods. Tomorrow’s temptation spending is not valued bytoday’s self, hence the tax.

5

2 Model

2.1 The Setup

The model presented in this paper is a modified discounted utility model (ie. so u(c0)+δu(c1) oru(c0) + β

∑t δtuct) with two separable utilities: U is the utility received from non-temptation

goods and V is the utility received from temptation goods. Both U and V are increasingand concave functions with the additional assumption that U is three times differentiable.For notational convenience, he separates non-temptation goods, such as a refrigerator, into xand temptation goods, such as donuts, into z; thereby giving us a one period utility functionU(x) + V (Z). Moreover, let c stand for the total amount of consumption, ergo: x+ z = c andthat x, z ≥ 0. Finally, Banerjee oscillates between using a two period and a three period modelfor illustrative purposes but the primary model entails only two periods, t = 1, 2. Incorporatingthese aspects we get:

U(x1) + V (z1) + δE(U(x2)) (1)

where the subscripts designate the period and E()̇ is the expectation function since heincludes random shocks in this exhibit of the illustrative model. Note that only the self inperiod 1 does values temptation goods so for period 2 self, V (z2) = 0. For this model, pricesare normalized to 1.

2.1.1 Period 1

In the first period the agent maximizes equation 1 subject to income in period 1, y1, thatcan be spent and/or saved; consequently the amount saved or resulting wealth in period 1 isy1−x1−z1 = w1 which can be invested into a future income generating function f(w1, θ) whereθ is a parameter for random shocks. Let f(,̇)̇ be an increasing, concave, and differentiablefunction.

2.1.2 Period 2

In the second period, the agent also received a labor income y2 but that income is uncertain solet y2 = y2(θ

′) where θ′ is a random shock variable that is independent of θ and θ′ occurs in thesecond period but before the agent must confront the maximization problem. Consequently,the agent’s total budget constraint is f(w1, θ) + y2(θ

′) which we set to c2 where c2 = x2 +z2. Combining the budget constraint and writing x2 as a function thereof we get x2(c2) =x2(f(w2, θ)+y2(θ

′)) and consequently let z2 be a function of x2 or z2(x2) subject to the constraintthat V ′(x2) = U ′(x2). Finally, the agent maximizes U(x2) + V (z2) where x2 and z2 are goodsin the second period.

2.2 The Maximization

Combining the elements from above into equation 1, in the first period the agent faces theensuing maximization problem:

U(x1) + V (z1) + δE(U(x2)) =⇒

U(x1) + V (z1) + δEθ,θ′ [U(x2(f(w1, θ)) + y2(θ′)] (2)

subject tow1 = y1 − x1 − z1 and x1, z1 ≥ 0

6

This above implies the Lagrangian:

L = argx1,z1,x2 maxU(x1) + V (z1) + δEθ,θ′ [U(x2(f(w1, θ)) + y2(θ′)] + λ(y1 − x1 − z1 −w1) (3)

Supposing that all necessary derivatives and an interior optimum exists, we get FOCs:

i. dL(.)dx1

= dU(x1)dx1

− λ = 0 =⇒ λ = dU(x1)dx1

ii. dL(.)dz1

= dV (z2)dz2

− λ = 0 =⇒ λ = dV (z1)dz1

iii. dL(.)dx2

= 0 =⇒ δEθ,θ′d(U(x2(f(w1,θ))+y2(θ′))

dx2

dx2(f(w1,θ)+y2(θ′)df(w1,θ)

df(w1,θ)dw1

= λ

iv. w2 = f(w1, θ) + y1 − x1 − z1.

Solving the system above by setting the λs equal gives us

dU(x1)

dx1=dV (z1)

dz1= δEθ,θ′

d(U(x2(f(w1, θ)) + y2(θ′))

dx2

dx2(f(w1, θ) + y2(θ′)

df(w1, θ)

df(w1, θ)

dw1

Now, given that in period two the budget constrain is c2(θ, θ′) = f(w1, θ) + y2(θ

′), we canplug c2 into the right-most expression to get something marginally less painful on the eyes:

δEθ,θ′d(U(x2(c2))

dx2

dx2(c2)

dc2

f(w1, θ)

dw1=dU(x1)

dx1(4)

Equation 4 shall, from hereon, be referred to as the Modified Euler Equation where the mod-ification from the standard Euler equation stems from the additional term dx2(c2)

dc2in Equation

4. Note that because U and f are non-negative by assumption and given that dx2(c2)dc2

≤ 1, theestimates of δ given the standard Euler equation will always be lower than the estimates givenby our equation 4 or the Modified Euler Equation. The authors attribute this to a temptationtax and call the aforementioned additional term dropped utility. The result can be interpretedas follows, the sophisticated agent knows that she cannot commit fully to satisfying her desiresfor x2 because z2 will tempt her to spend only a fraction of c2 on x2. Consequently, she incor-porates this tax into her maximization in period 1 resulting in a δ that is less forward-lookingthan the alternative we would observe in the standard Euler Equation without the temptationgood.

Finally, note that z(c)+x(c) = c =⇒ dz(c)d(c) + dx(c)

dc = 1 =⇒ dx(c)dc = 1− dz(c)

dc =⇒ dx2(c2)dc2

=

1− dzc(c2)dc2

. The paper uses this result to write Equation 4 as

δEθ,θ′d(U(x2(c2))

dx2

f(w1, θ)

dw1[1− z′(c2)] =

dU(x1)

dx1(5)

where the latter part is more readily interpreted as the temptation tax.

Standard Euler Equation Modified Euler Equation

δEθ,θ′d(U(x2(c2))

dx2

f(w1,θ)dw1

= . . . δEθ,θ′d(U(x2(c2))

dx2

dx2(c2)dc2

f(w1,θ)dw1

= . . .

”” δEθ,θ′d(U(x2(c2))

dx2

f(w1,θ)dw1

[1− z′(c2)] = . . .

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2.3 Generalization

The above framework was presented with only two goods x2 and z2 and solely for two periods,t = 1, 2 in name of better exposition. It can be trivially generalized through the exact samesteps for any finite bundle of goods that can be separated into temptation and non-temptationgoods and any number of future periods. For example, if we let {x1, x2, ...xm, xm+1...xn} bethe total bundle of consumable goods and let x = {x1...xm} and z = {xm+1...xn} under theassumption that our bundle has been divided appropriately then the framework above holds,just now with vectors x and z. Additionally, we can linearly combine any number of periodsthrough

∑Tt=s δt−sU(xt) + V (zs) whose maximization will yield the exact same intertemporal

equation as equation 4 but now with s+ 1 and s instead of 2 and 1 with the respective vectors.

3 Implications of the Modified Euler Equation

3.1 Demand for Durables

Let all of the above assumptions above the utility functions U and V hold as before. Let xbe a non-durable good and y be a durable good, where both x, y ∈ {non-temptation goods}.Furthermore, let y pay a separable utility ud in periods 1 and 2 (ie. U(x1) + V (z1) + ud +δ(U(x2)+ud) = U(x1)+V (z1)+δU(x2)+(1+δ)ud). Since the prices are relative to each other,the agent will be willing to pay ud

u′(x1)(1+δ)for x1 whereas the decision to save w1 will be based,

as before, on δ(1− z′(ct)). Consequently, if we are to only observe people’s discount rates fromthe purchase of durable goods, the discount rate will appear higher (ie: more patient agent)than otherwise. Moreover, this indicates that individuals who otherwise succumb to temptationgoods and are not disposed to saving will purchase durable goods as a commitment device tothe consumption of x goods.

3.2 Demand for Commitment

As mentioned in the paper, prior research has demonstrated that people demand commitmentdevices as a way of controlling their temptation. One such exhibit includes SEED accountsand was illustrated by Ashraf, Karlan, and Yin (2004). SEED accounts allow individuals withexisting bank accounts to put in money that they cannot withdraw until a particular date orgoal. This SEED money is intended by the agent to be used on non-temptation goods, the way

to incorporate it into the model is to have U ′(xt)V ′(z) especially high on the preset SEED dates or

goals.

3.3 ’sin’ taxes

Recall that Period 1 wants the Period 2 self to choose accorrding to the standard Euler Equa-tion SE whereas Period 2 will choose based on the modified Euler Equation ME. From ourdiscussion above, we know that SE ≥ME and the smaller dx2(c2)

dc2is the larger is the difference

between the two selves. Upon this basis, Banerjee examines the effect of sin taxes on temptationgoods or z goods, such as alcohol or cigarettes, implemented in period 2 on savings (ie dx2(c2)

dc2).

Assuming the price of x goods is still 1, let p denote the price of z in terms of x. Whereasbefore our budget constraint was x2 + z2 = c2 now we have x2 + pz2 = c2 to account for thenon-unity price of z2. So now, period 2 has the objective function:

U(x2) + V (z2) + λ(x2 + pz2 − c2)

Maximizing this is trivial, just differentiate w.r.t. x2, z2, λ and solve for λ to get:

8

pdU(x2)

dx2=dV (z2)

dz2

and now differentiate this equation with respect to c2 (use chain rule) to get

pU ′′(x2)dx2dc2

= V ′′(z2)dz2dc2

(6)

Now, differentiate the budget constraint x2 + pz2 = c2 with respect to c2 to get

dx2dc2

+dz2dc2

= 1 (7)

Solve Equation 6 for dz2dc2

and plug that into Equation 7 to get:

dx2dc2

+ p2U ′′(x2)

V ′′(z2)

dx2dc2

= 1

The above beauty can now be solved for dx2c2

which is what we were seeking by some easyalgebra. Namely,

dx2dz2

=1

1 + p2U ′′(x2)V ′′(z2)

=1

1 + RuRV

pU ′(x2)V ′(z2)

pz2x2

(8)

However, to simplify Equation 81 more, recall that the price of z goods is in terms of x; thereforepU ′(x2)V ′(z2)

= 1 and recall that x2 = c2 − pz2. Combine these facts to get:

dx2dc2

=1

1 + RuRv

pz2c−pz2

where RU and RV are the relative risk-aversion coefficients. In line with the footnote, let Uand V be CRRA functions so that p only enters through pz

c−pz2 = pzx in the above equation.

Consequently, as p increases, all else constant, dx2dz2decreases and since we are equating dx2

dz2with

savings, it suggests that an implementation or increase of sin taxes can actually decrease anagent’s savings as long as pz2 increases as p increases.

The conclusion is that sin taxes, although designed to deter the consumption of temptationgoods, may affect an agent’s savings behaviour negatively by increasing the amount of income anagent spends on temptation goods. Of course, decreased consumption thereof is also a possibilitybut the model presented here cannot account for that due to the absence of accountability forthe long run.

4 The Shape of Temptation

This section attempts to capture the essence of this Banerjee and Mullainathan paper, as thetitle suggests, by describing the consequences of varying z(c) or the shape of temptation. Inpractical terms, it can take any desired shape given that z(c) ∈ [0, C]. The result is shownbelow:

1You may ask, how do the last two equations come about? Check out this link: http://www.uio.no/studier/emner/sv/oekonomi/ECON4310/h11/undervisningsmateriale/CRRAutility.pdf for some information on CRRAutility functions. You essentially, differentiate CRRA for U and V twice and substitute that result into ourderivation of dx2

dz2.

9

Proposition 1 Assume that the U function is known and fixed. Let z(c) and x(c) be a pair ofnon-negative valued, strictly increasing functions defined on c ∈ [0, C] for some C > 0 such thatz(c) + x(c) = c. Then there exists an increasing, differentiable and strictly concave functionV defined on [0, z(c)] s.t. the assumes z(c) and x(c) functions are the result of maximizingU(x) + V (z) subject to a budget constraint x+ z = c and the conditions x, z ≥ 0.

Proof: Assume that U is a concave, strictly increasing function (the paper does not quitemention this but I believe it is implied because otherwise contradictions amass). Becausez(c) is strictly monotone on a closed interval, we know z−1(c) exists (recall z(c) = z =⇒z−1(z) = c because z(c) is bijective). Let g(z) = x(z−1(z)) and let V (z) =

∫ z0 U

′(g(y))dy.By the Fundamental Theorem of Calculus, it follows that V ′(z) = U ′(g(z)) and that V (z) isdifferentiable. Additionally, since z(c) is strictly increasing, z′(c) > 0 =⇒ (z−1)′(c) > 0 fromwhich it follows that U ′(g(z)) > 0 =⇒ V ′(z) > 0 and since U ′(c) is concave and increasing itfollows that V ′(z) is also concave and increasing.

Since no assumptions were made on the second derivative of V , we do not know whether V isweakly or strictly concave. The paper spends its remaining pages on exploring the consequencesof this difference. Namely, he explores the consequences of z′(c) being constant or z′′(c) < 0 ora decreasing z′(c).

Constant z′(c) Decreasing z′(c)dzdc is constant d2z

dc2< 0

As consumption(or income) goes up, tempta-tions maintains a proportion of total consump-tion. This includes no tempation (ie: z2(c2) =0)

As consumption(or income) goes up, tempta-tions become a smaller proportion of totalconsumption.

4.1 Declining (DTC) and Non-Declining (NDTC) Temptations

Recall Equation 5 or the Modified Euler Equation where 1− z′(c2) denotes the temptation taxthe self in period 2 imposes on the period 1 self. From in the above Table 4, let the constantdzdc case be NDTC and declining case be DTC. The latter case implies that V is more concavethan U (ie: V ′′(z) ≤ U ′′(x) ≤ 0), which is illustrated in the Appendix under Proposition 2. Inhuman terms, this implies that temptation goods are more tempting than x goods. The papercites one possible contradiction in the form of people with near 0 consumption (ie: people onthe verge of starvation) in which case the consumption of a nutritious food x is likely to be morevaluable than a donut or z. This, of course, can be duked out but for brevity, lets disregardsuch cases and focus on states where this doesn’t hold.

Justifications for temptation goods being more tempting than x goods include:

1. Temptations are often visceral and guided by our evolutionary physiology which can beobserved in our innate cravings for things such as sex, sweets, fatty foods, etc. If I, forexample, crave a candy bar I buy, eat, and I’m sated. Buying 10 more candy bars doesnot make much sense for my tempted me so I is difficult to expend a lot without reachingsatiation.

2. It is more difficult to satisfy a utility for x. For example, a house, good x cannot beconsumed in small amounts and requires a large financial input. For example, I may beexpend a lot of effort and set aside substantial money for a house but not reach satiationsimply because I have yet to save enough to actually sate the desire.

10

3. Temptation goods can be consumed immediately so they are impulsive. In other words,one doesn’t usually stand in the candy aisle pondering on how that Snickers bar will alterone’s path in life. On the other hand, buying house is a process so even with sufficientfinancial resources it inherently involves more discussion and contemplation through thearrangement of credit, discussion with lawyers (which are also born costs) etc so x goodsare typically not impulsive.

4. There are empirical studies, such as ones done by Deaton and Subramanian(1996) andBanerjee and Duflo(2007) that suggest declining temptations in increasing income.

4.2 Relation to Hyperbolic Discounting

The authors also extend or re-interpret the model from the Hyperbolic Discounting Modelperspective. Recall the hyperbolic discounting model:

U(xt) + βT∑s=1

δsU(xt+s)

4.2.1 Hyperbolic Discounting with two heterogeneous goods

Now, take the above model and add in V (zt) to get the part that the self in Period 1 maximizes:[U(xt) + βx

T∑s=1

δsU(xt+s)

]+

[V (zt) + βz

T∑s=1

δsV (xt+s)

]Note that in the original model, for s > 0 the V (zt+s) = 0 but here subsequent periods alsocontain V (z). We do not have an explanation for this, it could be an error or a way of param-eterizing for the effect of hyperbolic discounting on temptation goods.

For illustrative purposes, consider only the three period model:Period 1 maximizes:

U(x1) + βxδU(x2) + βxδ2U(x3) + V (z1) + βzδV (z2) + βzδ

2V (z3)

Period 2 maximizesU(x2) + βxδU(x3) + V (z1) + βzδV (z2)

Here we have two sources of inconsistencies. The first, particular to the paper’s model,stems from the fact that Period 1 self does not value V (z2) and U(z2) the same which can beseen from the fact that we assume βx 6= βz. Period 2, on the other hand, weighs V (x2) andU(x2) the same which can be seen from the absence of the β parameter.

The second inconsistency stems from the regular time inconsistency of the hyperbolic model:Period 1 discounts periods 2 & 3 by βδ and βδ2 so the only difference is between δ and δ2. Onthe other hand, Period 2 discounts period 3 at βδ < δ so Period 2 values Period 3 less thanPeriod 1 values Period 3.

4.2.2 Hyperbolic Discounting with two homogenous goods

An alternative way of building a Hyperbolic Discounting Model is instead of assuming tempta-tion and non-temptation goods, zt and xt respectively, we only assume ct; thus some goods arenot more tempting than others. In such a case, the agent maximizes: U(c1)+βδU(c2)+βδ2U(c3)s.t. c1 + c2 + c3 = w

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Now, without delving into the optimization because the result is, ostensibly, well known andcan be seen at detail in Harris-Laibson(2001), the optimal for the Period 1 self is:

U ′(c1) = βδU ′(c2)[1− βc2(w2)] (9)

where c2(w − c1) is given by Period 2 self’s optimum: U ′(c2(w2)) = βδU ′(w2 − c2). Note thesemblance between Equations 5 and 9, the temptation tax is captured in both equations in thesame manner.

Moreover, as seen in the paper’s fully coherent derivation on page 19, c′′2(w2) < 0 ⇐⇒U ′′′(c3)

[U ′′′(c3)]2< βδU ′′′(c2)

[U ′′′(c2)]2. With the LHS holding as an additional condition on the model, we get

the same result on the RHS as before, namely that V is more concave than U . However, due tothe difficulty of interpreting this and empirically testing it, this specification of the HyperbolicDiscounting Model is not as conducive to the exploration of the shape of temptation.

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5 Implications of DTC and NDTC

13

Consumption Smoothing

One of the most robust predictions of the standard model of savings is that an increase in future earnings (𝑦2) will reduce today’s savings and increase today’s consumption (𝑐1). This is the direct result of the desire for consumption smoothing, induced by diminishing marginal utility. An increase in future income generates a desire to spend more both today and tomorrow according to income effect.

𝑑𝑐1

𝑑𝑦2> 0

If temptations are declining then there exist utility functions for which there is a range of (𝑦2), where consumption in period 1 decreases with an increase in income in period 2,

𝑑𝑐1

𝑑𝑦2< 0

It tells us that those who are sufficiently poor might actually react to the prospect of future income growth by beginning to save more since their temptations will decline as their income increases. This offers a possible interpretation of the idea that aspirations matter (Ray, (2002))

Intuitively, when you are poor and have a very small amount of income today, when your future income increases you know that if you consume less temptation goods today and save for tomorrow, the temptation tax you will face tomorrow will be less (Due to declining temptation case when your consumption (income increases so your consumption will also increase) increases fraction of the marginal dollar spent on temptation goods will decrease). So you have more incentives to save for tomorrow today since you know that a smaller portion of your income will be deducted as a temptation tax due to declining temptation assumption.

Poverty Trap

Poverty Trap describes a situation in which poor people get trapped in poverty and be

unable to escape it. In pervious literature, poverty trap is traditionally explained by

limited access to credit and capital markets, extreme environmental degradation (which

depletes agricultural production potential), corrupt governance and etc. However, this

paper states that a diminishing temptation alone can generate a behavioral poverty trap.

Implication's of Declining TemptationsContent for this section is brought to you by

2014's Debreau team

Although we are not allowed to replicate the thorough proof of this statement in this note,

due to the limited space, we can provide the basic intuition of it as follows.

The richer individuals have more initial income/wealth (which in this model are the same

thing) , 𝑦1, to start out, so they are able to leave more to future selves, i.e. 𝑤1 is higher.

Since 𝑓(𝑤1, 𝜃) is an increasing function of 𝑤1 and 𝑐2 = 𝑓(𝑤1, 𝜃) + 𝑦2(𝜃′), the second

period consumption will be higher for wealthier people. In a word, 𝑐2 is an increasing

function of 𝑦1.

Proposition 4 Assume that there are no shocks i.e. that both 𝜃 and 𝜃′ are constants. Then

as long as we are in NDTC, 𝑐2 will be a continuous function of 𝑦1. On the other hand in

DTC, there may exist a �̅�1, 0 < �̅�1 < ∞ such that 𝑐2 jumps discontinuously upward �̅�1.

The proposition above is actually saying that there is a critical wealth level, �̅�1 such that

second period consumption, 𝑐2, discontinuously jumps up about �̅�1, as the figure above

shows.

So under the assumption of decline temptation, those individuals with initial wealth

above �̅�1 can consume much more in the second period than those poor people with

initial wealth level just below �̅�1.

Furthermore, declining temptation implies decreasing z′(𝑐2), which indicates that the

“tax” charged on rich people is significantly lower than that on the poor. So rich

individuals are more motivated to save, or say investment generally, then become even

wealthier because of the profit from the investment. While poor people save very less or

just dissave, then persist in poverty trap generation by generation.

Demand for Credit

𝑦1̅̅ ̅𝑦1

𝑐2

In declining temptation

cases

Declining temptations also have implications for the benefits and costs of credit. In the

traditional model, access to credit is clearly good: it increases the opportunity set. In

models with self-control problems, credit can potentially hurt. Specifically, today’s self

could be made worse off if tomorrow’s self has access to credit. This general feature of

self-control models has more specific implications if we focus on declining temptations.

To fully capture commitment benefits in this context, we need to introduce a third (zero)

period before the existing two-period model.

Assumptions:

1. The zero period self has a decision: whether or not to allow the period one self to

have access to a loan of type λ, by which the period 1 self can move consumption

from period 2 to period 1.

2. A loan of type λ is defined by two characteristics: 𝑟(𝜆), the interest rate and a

maximum loan size 𝐿(𝜆). If the period one self is allowed a loan of type λ, he can

choose to borrow some amount between 0 and 𝐿(𝜆) and have the period two self

repay 𝑟(𝜆) times that amount.

From the assumptions above, we can see why we need period zero. If there are only two

periods, the optimal strategy for period one self is definitely making no commitments on

tomorrow’s self consumption at all, in other words, letting him use up all the resources,

since it is the last period. The only chance that the individual can be better off from

making commitments is that we let him capable of constraints his tomorrow’s spending

to x, rather than z. However, we only talk about commitments on total resources here.

Therefore, we add the period zero at the beginning so that we can discuss the cost and

benefits of commitments.

Since we are interested only in the investment and commitment demands of this period

zero self, we assume this self has no consumption. Instead he or she merely maximizes

U(x1) + δU(x2)Proposition 8.1 Under NDTC, if the period zero self is willing to allow a loan λ ={r(λ), L(λ) }, he will always willing to allow loan λ′ = {r(λ′), L(λ′) } as long as r(λ) = r(λ′) and L(λ′) ≤ L(λ)

The detailed proof can be found in the original paper. Generally, proposition 8.1 indicates

that when temptations are constant or decreasing, one-period self only has an upper

bound for credits. If a certain amount of loan is permitted, any loan of smaller amount at

the same interest rate should also be permitted.

Intuitively, period zero self concerns about over-borrowing in NDTC cases, because it

hurts the period zero self in two ways: (i) exaggerating the difference in consumption

between the period one and two selves and (ii) Engaging in borrowing at a higher rate

than the period zero self would want (because the period one self actually discount his

tomorrow’s assumption at rate of δ(1 − z′(c)), but period zero self assumes it to be δ).

Proposition 8.2 Under DTC there will exists situations where he is willing to allow a

loan λ = {r(λ), L(λ) }, but not a loan λ′ = {r(λ′), L(λ′) } where r(λ) = r(λ′) and L(λ′) < L(λ).

Again, we don’t want to exhaust the readers with the original complex proof, so we

provide a more intuitive but not that rigor one instead.

Since the declining temptation indicates that V(z) is more concave than U(x), so it is

reasonable to assume them as the following graph shows,

Specifically, V(z) is assumed to be weakly concave and kink at 𝑐̅. As the figure shows,

for consumption level less than 𝑐̅, the marginal utility of consuming temptation z is

higher than that of x. So if y < 𝑐̅, the individual will spend all of his or her money on

temptations.

So in the case that 𝑦1 = 𝛼𝑐̅ (0 ≤ α ≤ 1), 𝑦2 = 𝑐̅ + k (for simplicity, the paper assumes

that 𝑓(𝑤1, 𝜃) = 𝑤1, no profit from savings), if there is no borrowing allowed, the utility

of zero period self Π0 is

Π0 = δU(k) Now assume that there exists a loan λ = (1, L(λ)), where L(λ) < (1 − 𝛼)𝑐̅ < 𝑘, so

𝑦1 + L(λ) < 𝑐,̅

one period self would still spend all his money on temptation z, and for two period self,

𝑦2 − L(λ) = 𝑐̅ + 𝑘 − L(λ) > 𝑐̅So the utility of zero period self becomes

Π0′ = δU(k − L(λ)) < Π0

Therefore, overly small loan will hurt the utility of one period self, individuals also have

a lower bound for loan size in DTC cases.

These results are helpful in helping us parse credit contracts that are observed. Consider

two different contracts: micro-finance and credit cards. Micro-finance offers only larger

loans while credit cards only offer small loans. In the model with constant temptations,

and no lumpy investment, the period zero self may accept to get a credit card but refuse a

micro-finance loan, but never the other way around. Declining temptations can explain

why we may observe the opposite: resistance to credit cards (or the equivalent) while

support for micro-credit.

U(x)

V(z)

𝑐̅

Supply for Credit

So far we have focused on the demand side: We now turn to the supply side of credit and

investment opportunities. Consider an individual who is borrowing from a monopolistic

money lender. Suppose a new investment arises that earns a rate of return higher than the

money lenders cost of capital. If the money lender could price discriminate and offer a

rate specifically for this investment, would the investment get made? Simple Coasian

logic suggests it should. Yet there is a long tradition of arguing that the money lender has

an incentive to block this investment either directly or by refusing to finance it, in order

to maintain his power over the borrower (Bhaduri 1973). The logic is that the

technological improvement will raise the earnings of those who used to borrow from the

money-lender which in turn hurts the lenders profits through reduced borrowing. In short,

the money-lender prefers to have his clients caught in a debt trap.

In the this sub-section we can see that while the intuition proposed in the previous

paragraph continues to hold under NDTC, the possibility of declining temptations

reintroduces the possibility that Bhaduri had emphasized: The money-lender may indeed

want block progress to keep the borrower in his thrall.

Proposition 9 In this setting under NDTC, the money lender will always be willing to

lend the investor money to make the new investment possible. Under DTC there exist

situations where this is not true.

6 Empirical Tests of the Model and Extensions

Among the most cited papers that cite this one, we were only able to find one theoreticalpaper that uses decreasing temptations. Cassar and Wydick (2012) provide a short, 7-pagepaper that incorporates decreasing temptations into the famous credit market model of Stiglitzand Weiss (1981). Stiglitz and Weiss involves asymmetric information between borrowers andlenders creating a moral hazard in which borrowers to have incentives to invest in risky projects.By incorporating declining temptations, Cassar and Wydick find that present bias, rather thanincentives, is the primary mechanism behind risky investments under asymmetric informationin credit markets.

As Banerjee and Duflo (2010) say, development economics has relatively recently reemergedinto the mainstream of economics departments (for a history of ”Development Economics”separating itself from the mainstream economics departments and into policy organizationssuch as the World Bank, see Easterly (2014)). They note that it has become particularlypopular in this field to tightly integrate theoretical work and empirical testing. The virtues offield experiments are emphasized. The developing world, they say, ”serves as a testing groundfor fundamental economic theories and the source of exciting new ideas.”

Since this paper was recently written and unpublished, it makes sense that researchers wouldseek experiments that test the assumptions and implications of this model before expanding thetheoretical framework. For instance, Banerjee and Duflo (2010) discuss how the results of thispaper are consistent with Banerjee, Duflo, Glennerster, and Kinnan (2009), a study that findspositive effects from microfinance programs. They speculate that microfinance is a way for thepoor to help themselves commit to a savings plan that mitigates the impact of temptations.Another example of this theory being tested is Spears (2011), who uses a laboratory experimentand field experiment (in India) along with the American Time Use Survey to test three potentialtheoretical mechanisms of poverty. Spears isolates a causal effect of poverty on behavior, but isunable to separate the effects of the mechanisms. Below is a graphic from Spears that presentsthe mechanisms.

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Within the paper itself, Banerjee and Mullainathan discuss potential empirical tests ofthe model. One is using exogenous income shocks to test income elasticities and differencesbetween the rich and the poor. They also suggest experimental frameworks where the decliningtemptation assumption is itself is tested. The last thing they suggest a framework to test iswhether or not the differences in time preference between the poor are genuine, or a matterof composition as suggested by the model. Implicit in these proposed tests is a determinationwhat actually constitutes a temptation good.

It is worth mentioning that section 5 of this paper presents implications of the model thatcan reasonably be thought of as theoretical extensions. These are expanded on in other sectionsof this handout.

In their conclusion, Banerjee and Mullainathan suggest other potential extensions. Oneuse they see for the model is helping to understand and craft savings products that help com-mitment. Features that could be explored are mechanisms like forcing savings and limitingwithdrawals. Another potential extension is using it beyond pure expenditure decisions andfor decisions like labor leisure choice. Thirdly, they point out that the model generates a non-convexity in the second period from the perspective of the first period self. They suggest it canhelp in a utility function that was proposed by Friedman and Savage (1948) to explain decisionsto do things like purchase lottery tickets. They also see potential implications for the relationbetween measured and actual risk more generally.

There are also theoretical extensions we suggest. One potential way the idea of temptationgoods and declining temptations can be used is in non-monetary contexts such as time allocation.This may be similar to what the authors had in mind when they suggested focusing on laborleisure choice. The period one self could allocate their resource, time, to getting a productivetask done during period two, assigning zero utility to potentially tempting distractions. But thenthe period two self may gain present utility from those distractions. We think many studentswouldn’t hesitate to classify time spent on Facebook that was originally meant for workingon an Econ 605 presentation as a temptation good. And this leads us to the question: canthe wealthy better afford wasting this time? This could introduce interesting questions abouta potentially offsetting effect between declining temptations and opportunity costs, which arehigher for the wealthy. See the Critique section for another potential extension we propose tothe utility function.

7 Conclusion

The authors see the importance of their paper lying in the fact that the structure of temptations,which in practical terms are ”wasted expenditures” are key to understanding the poor. Theyask about what they view as an important feature of this structure: whether the impact oftemptations diminishes with income. They lay out further questions that need to be answered,which are explored in the ”Empirical Tests and Extensions” section of this handout. Theyacknowledge future work needs to be done, but they have provided an extremely insightfulframework. As of Friday, January 23, 2015 it has 140 citations according to Google Scholar-which is remarkable for a working paper.

8 Critique

For many examples, the assumption that the rich and the poor consume similar amounts oftemptation goods, but the rich can better afford these temptations seems reasonable. Forinstance, white Americans, who are on average significantly wealthier than black Americans,

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use illegal drugs at a similar rate to black Americans. However, for some temptation goods,the assumption is not as consistent with the data. For example, the poor smoke cigarettes ata much higher rate and at a higher absolute quantity than the wealthy. Additionally, peoplewith mental health concerns (even ones that could be considered ”minor”) smoke more. Mentalhealth is plausibly related to myopic behavior. It is not the purpose of this handout to discusswhich way the causal arrow points between wealth and cigarette smoking. The point is merelythat -at least in America-smoking rates are not consistent with declining temptations.

The utility structure of temptation goods is also worth reflecting on. Introspection tells usthat temptations are not only goods that have zero utility before they are used and positiveutility as they are used, but also goods that have negative utility after they are used. InEnglish, people often regret it after they give in to temptations. Perhaps negative utility inthe period after the one in which the temptation good is used serves as a learning mechanismin certain contexts. It is also feasible that this learning mechanism is heterogeneous amongdifferent individuals. The implications of this structure would have to be explored formally toadequately discuss them. This can also be seen as a potential extension.

Additionally, it is important to realize that it is implausible that declining temptations isthe only mechanism that can explain perceived myopia. This is not to suggest that the authorsimply this, but that a sloppy reader may infer it. Mullainathan agrees. Karlan, McConnell,Mullainathan, and Zinman (2009) -directly after mentioning the model in this paper -say, ”Wedevelop some theory and evidence that a different psychology - limited attention - also plays animportant role in explaining these types of intertemporal choices.” It is important to rememberthat good models can tell us about isolated aspects of reality-not provide a full explanation ofit. To quote George Box, ”all models are wrong, but some are useful.”

Another ill-conceived usage of the results presented in this paper is to say that it proves thepoor are not more myopic than wealthier individuals. The authors don’t suggest this. However,like the previous example the results can be misused in this way. To see this, only introspectionis required. It doesn’t seem at all plausible that there is no relation between genuine myopiaand financial success. We make no judgment as the to degrees with myopia is a result ofdisposition, environment, and/or free will, yet it seems obvious that people have different levelsof actual myopia. And all other things equal, you would think that genuinely myopic peoplewould make less money than genuinely disciplined people. We ask you, the reader, to reflect onyour own life, and consider times when you behaved more myopically and times you were moredisciplined. Which of those times have been more instrumental to whatever degree of academicor financial success you currently have? Someone may respond to this criticism by arguingthat introspection is unscientific. But when talking about psychological assumptions, when isintrospection not used to a large extent? These assumptions must originate from somewhere.The great theorist Ariel Rubinstein once said, ”Introspection, which I believe is the main tool ofexperimental economics or the cognitive psychologist is something that overall this is the mostimportant tool in the field.”

References

[1] Banerjee, Abhijit V., and Esther Duflo. Giving Credit Where It Is Due. The Journal ofEconomic Perspectives 24, no. 3 (Summer 2010): 61-79.

[2] Banerjee, Abhijit, Esther Duflo, Rachel Glennerster, and Cynthia Rinnan. 2009. The Miracleof Microfinance? Evidence from Randomized Evaluation.

[3] Cassar, Alessandra and Bruce Wydick, Credit Rationing with Behavioral Foundations: Re-visiting Stiglitz and Weiss (2012).Economics. Paper 27

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[4] Easterly, William. The Tyranny of Experts: Economists, Dictators, and the Forgotten Rightsof the Poor. New York: Basic Books, 2014.

[5] Friedman, Milton and Leonard Savage. Utility Analysis of Choices Involving Risk. Journalof Political Economy 56, no. 4, August 1948, 279-304.

[6] Spears, Dean. ”Economic Decision-Making in Poverty Depletes Behavioral Control.” TheB.E. Journal of Economic Analysis & Policy 11, no. 1 (2011).

[7] Stiglitz, J.E. & Weiss, A. 1981, Credit Rationing in Markets with Imperfect Information,American Economic Review, vol. 71, no. 3, pp. 393-410.

[8] Lopez, German. Everyone does drugs, but only minorities are punished for it. Vox.

[9] Pollack, Harold. Tobacco is still America’s top health threat. But Washington doesn’t treatit that way. Washington Post.

[10] Rubinstein, Ariel. ”Rubinstein on Game Theory and Behavioral Economics.” EconTalk.2011. http://www.econtalk.org/archives/2011/04/rubinstein_on_g.html

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