Bad Habits and Endogenous Decision Points∗

Peter Landry†

March 20, 2014

Abstract

I present a theoretical model of addiction in which cravings are costly distractions

that force the individual to “think about” the associated consumption decision. By

allowing choices to influence the frequency of future cravings, the model emphasizes

new incentives and disincentives; namely, consumption provides a temporary break

from these so-called “decision points,” but increases their long-run frequency. Match-

ing evidence unaddressed by prevailing theories, raising present consumption reduces

near-term future demand by delaying the next consumption occasion, but leads to both

higher frequencies and higher per-occasion levels of consumption in the long-run. In-

corporating random external cues (modeled as decision points) into the cravings model,

consumption becomes more predictable and less cue-sensitive as habits strengthen. Al-

though cues generally boost demand, overexposure deters consumption, in line with

evidence on advertising “wearout.” Lastly, a method to elicit users’ “natural” deci-

sion points is proposed, which carries unique and testable implications regarding the

time-pattern of precommitment.

∗Philipp Sadowski and Peter Arcidiacono provided excellent guidance at different stages of thisproject. I also thank Attila Ambrus, Jonas Arias, Colin Camerer, Mike Dalton, Joe Hotz, RachelKranton, Mehmet Ozsoy, and Curt Taylor for helpful feedback.†Division of Humanities and Social Sciences, California Institute of Technology, MC 228-77, 1200

E. California Blvd, Pasadena, CA 91125. E-mail: plandry@caltech.edu.

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1 Introduction

The choices we make crucially depend on the decisions we consider — I cannot choose

to consume a product without considering the decision to consume it. What then

causes someone to consider a particular consumption decision at a particular moment

in time? Further, when faced with the decision, can one’s chosen action influence the

frequency with which this same decision will be considered in the future? Such ques-

tions have remained outside the scope of economic analysis because standard practice

takes as given when decisions are faced. In repeated choice models, it is custom-

ary to assume time is metronomic. An inherent feature of the standard discrete-time

(t=0, 1, 2, . . .) specification and its continuous-time limit, metronomic decision-making

entails an exogenously-fixed “decision schedule” in which decisions are evenly spread

across time — akin to a practicing musician who uses a clicking metronome to establish

a fixed tempo.

While often an empirical necessity, I claim that the rigidity of standard time is a

major theoretical limitation in the realm of habits.1,2 For instance, the development of a

habit is typically characterized by simultaneous rises in the consumption frequency and

in the level of consumption per occasion. Metronomic decision-making is not amenable

to accurate micro-level descriptions because, in a standard interior solution, the fixed

time-interval between decisions implies a fixed consumption frequency. Moreover, I

argue that standard time conceals the defining incentives (as well as an important

cost) of so-called bad habits, namely the incentive that drives addiction, a potentially

troublesome habit for utility theory since addictions seem to bring more harm than

benefit.

While exploring the themes described above, this paper develops a new framework

for bad habits based on “endogenous decision points.” Defined as the points in time

(t = t0, t1, . . .) an individual faces the decision of interest, the decision points are

endogenous in that they depend on past consumption. The basic model is built on

the concept of a craving, or more specifically, the idea that when a craving strikes, it

distracts its target from whatever else she happens to be doing and forces her to think

1Habits constitute a substantial portion of our everyday activities. From diary accounts of day-to-daybehavior, Verplanken and Wood (2006) estimate 45% of documented actions were habits. Other studiescontend most behavior is habitual (Louis and Sutton, 1991; Verplanken et al., 2005).

2Beyond the descriptive and normative limitations that are highlighted in this paper, metronomic decision-making precludes plausible comparisons of subjective day-by-day experiences since it implies that someonewith a full-strength consumption habit and someone who has never consumed the good will think about theactivity equally often. Whether the good in question is a video game, a drug, a fragrance, etc., it is unlikelythat someone who has never used the good thinks about it anywhere near as often as a frequent user.

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about the craved good. Thus, cravings are modeled as decision points.3 Furthermore,

because cravings are distracting, each decision point will entail an implicit opportunity

cost, so that the individual does not want to face the decision.

Motivated by properties of cravings, two core assumptions relate consumption choices

to the timing of future decision points:4 (i) by reinforcing the habit, consumption in-

creases the long-run decision point frequency; and (ii) consumption reduces the short-

run decision point frequency. From (i), a strong habit is “bad” because it involves

frequent unwanted decision points. However, the short-run effect in (ii) embeds a

strategic incentive to consume as a means to ward off imminent cravings, allowing

outside activities to proceed uninterrupted in the interim. This, I argue, is the crucial

incentive that sustains addiction.

As shown in Section 2, the cravings model generates a stable steady-state of ad-

diction with a “slippery slope” equilibrium path. The addict’s behavior is jointly de-

fined by the consumption frequency and the level of consumption per occasion, which

matches pharmacological evidence: as Benowitz (1991) describes, addiction involves

“certain rates of delivery and certain intervals between doses.” Both the frequency and

the per-occasion level of consumption grow on the path to addiction, which motivates

a dual measure of habit strength, while uniquely capturing evidence that cigarette

addicts inhale more nicotine per cigarette than occasional smokers (Shiffman, 1989;

Shiffman et al., 1994). As alluded to earlier, standard addiction and habit-formation

models do not feature simultaneously varying levels and frequencies of consumption.5

3The decision point is a natural formalization because upon the arrival of a craving, an addict becomesfixated on the object of dependence: as neuropsychology findings indicate, cravings reorient decision-makingfaculties towards the craved good, where the phrase “reorients decision-making faculties” refers to whatpsychologists may describe as enlistment of “cognitive control resources” or as prompting an “attentionalswitch.” Neuroimaging data show how drug cravings activate working memory systems (Grant et al., 1996;Garavan et al., 2000); as Garavan and Stout (2005) describe, “working memories are occupied with drugcraving thoughts and ruminations.”

4Although the opposing long- and short-run effects of consumption on future decision points may be es-pecially pronounced in the realm of addiction, (i) and (ii) appear to generalize to a broad class of behaviors.To start, (i) can be motivated as a general property of memory, where the more I’ve done something in thepast, the more likely (or often) I’ll think about doing it in the future. Meanwhile, (ii) reflects the notion thatif I’m thinking about doing something and then I do it, my mind will naturally move onto something else.While this paper will focus on drug addiction (cigarettes, in particular), other examples for which (i) and (ii)likely hold include: eating a peanut butter and jelly sandwich, visiting a particular website, drinking a Pepsi,checking email, buying an iPhone, watching a particular TV show, visiting a particular store, making yourbed (and other everyday chores), etc. As an illustration, if I’ve made my bed every day for the past severalyears, I’m more likely to think about making it tomorrow than I otherwise would be; that said, if I made mybed a few minutes ago, I’m less likely to consider making the bed than if it was still unmade.

5To be sure, discrete-time models with stochastic or otherwise time-varying influences could conceivablypermit consumption frequencies that vary with habit strength. However, in doing so, such models would: (i)unrealistically restrict all time-intervals between consecutive consumption occasions to be an integral multipleof some arbitrary period length; and (ii) fail to accommodate the stable and predictable consumption patternsof addicts, characterized by consistent doses separated by consistent intervals. Instead of the interpretation

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Section 3 establishes adjacent substitution, which means demand in the near-term

future falls with present consumption; i.e. the good now and in the near-term future are

substitutes, a reversal of its long-term correlation, “distant complementarity.” Consis-

tent with laboratory evidence (Zacny and Stitzer, 1985; Dallery et al., 2003), adjacent

substitution is interval-driven because the time-interval to the next consumption oc-

casion rises with current use. Adjacent substitution is noteworthy in the context of

addiction because its opposite — adjacent complementarity — is at the foundation of

the prevailing “rational addiction” model, introduced by Becker and Murphy (1988).

Moreover, through its interval-driven feature, adjacent substitution is relevant to the

puzzle of addiction since it exists as a manifestation of the underlying incentive to

consume.

To assess what conventional discrete-time analysis can and cannot tell us when

decision points depend on past consumption, Section 4 considers measurement and

inference when model-generated choices are aggregated to discrete-time. Although its

interval-driven feature is concealed, measured adjacent substitution distinguishes the

model from prevailing standard-time models (most notably, rational addiction the-

ory), provided the data aren’t too coarse. By showing how discrete-time inference

can skew the underlying time preferences, the model also reconciles several apparent

time-preference anomalies — commodity-dependent discounting, implausibly severe

impatience, and endogenous discounting — that are prominent in both theoretical and

empirical research on addiction.6 Furthermore, the exercise shows how the predomi-

nant, history-dependent “habit-formation preferences” may be an artifact of discrete-

time aggregation. This portrayal fits with the neuropsychology conclusion that drugs

are increasingly ‘wanted’ despite not being increasingly ‘liked’ in the course of addic-

tion in that demand grows even though static consumption preferences do not (e.g.

Robinson and Berridge, 1993, 1998, 2008).

In Section 5, external cues are modeled as random decision points and integrated

into the cravings framework. Consistent with evidence and in contrast to prevailing

cue-based addiction theories, the model shows how consumption patterns become more

predictable and less sensitive to external cues as an addiction takes hold. As the cue-

arrival rate increases, consumption frequencies rise (initially), but per-decision levels

that all consumption occurs at times t = 1, 2, . . ., as is explicitly stated in most theoretical models, somediscrete-time models are agnostic on the allocation of period-t consumption over the interval [t, t + 1), inwhich case fine-grain descriptions of consumption behavior are not within such models’ scope.

6Section 4.2 overviews these studies. As an example of the theoretical/empirical interplay in this domain,severe impatience helps justify harmfully addictive consumption in rational addiction theory and is also evidentin Becker et al.’s (1994) follow-up empirical analysis of rational addiction, which estimates implausibly highannual subjective discount rates, ranging from 56 to 223 percent.

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drop. In fact, the model predicts that overexposure to cues will wipe out an individual’s

demand, which fits with findings of “wearout” from overly-repeated advertisements.7

From the cues-as-decision points concept, a method to experimentally elicit a user’s

natural decision points is proposed in Section 6. In particular, the user specifies up

front the times that the experimenter will offer the good during the course of an exper-

iment, where such offers make the good momentarily available for consumption. The

chosen “availability schedule” in this scenario will in principle correspond to the user’s

anticipated decision points. Such decision-point elicitation could potentially alleviate

the limitations in measurement and in inference under discrete-time aggregation, as

covered in Section 4. Moreover, the offer-availability paradigm (and simple variations

of it) can be used to quantify other key features of the model, such as: (i) variation

in decision-point frequencies between addicts and those with weaker habits; (ii) con-

sumption’s short-run effect on future decision points; and (iii) the opportunity cost

of a single decision point.8 Lastly, the offer-availability paradigm gives rise to a new

prediction regarding the time-pattern of precommitment in that a user wants periodic

future access to the good, separated by periods with no access. In contrast, standard

rational addiction theory predicts that a user would want no restrictions on future ac-

cess, while present-biased addiction models predict that a user would want a complete

restriction.

As indicated, the decision points model will be repeatedly compared to Becker

and Murphy’s (1988) rational addiction theory as well as its subsequent adaptations,

such as Orphanides and Zervos’ relaxation of perfect information (1995) and endog-

enization of time-preferences (1998), and Gruber and Koszegi’s (2001) integration of

quasi-hyperbolic discounting. In a related theory, Gul and Pesendorfer (2007) model

addiction through a utility function that depends on yesterday’s choice and today’s

menu. For the cues extension of the cravings model, the two main benchmarks will be

Laibson’s theory (2001), in which cues affect consumption preferences, and Bernheim

and Rangel’s (2004) theory, in which cues trigger “mistaken” consumption. Empirical

evidence on addiction will be cited where appropriate. The bulk of cited evidence per-

tains to cigarette use because smoking offers an intuitive illustration of the model and

because cigarette addiction is unrivaled in its scale.9

7See Cacioppo and Petty (1979), Calder and Sternthal (1980), and Pechmann and Stewart (1989).8In turn, this estimated “distraction cost” can be used with measurements from (i) and (ii) to quantify

(respectively) the cumulative distraction costs of an addiction and the incentive to consume as a means toreduce the near-term decision-point frequency.

9There are one billion smokers who smoke nearly six trillion cigarettes a year, sustaining an industry thatcollects nearly a half-trillion dollars in annual revenues from a product projected to kill one billion peopleduring the 21st century (Eriksen et al., 2012).

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2 Basic Model: Cravings as Decision Points

An individual faces a repeated decision indexed by i = 0, 1, 2, . . .. The decision points,

(t0, t1, . . .), are the times the decision is faced. Decision points interrupt some outside

opportunity, so each ti carries an implied opportunity cost. This so-called decision

opportunity cost is normalized to −1 for simplicity, so its time-profile looks like this:

. . . . . .

. . .

−1

0ti−1 ti ti+1

t

At a decision point ti, the individual chooses a consumption level ci ∈ [0, 1] and

receives instantaneous utility u(ci). We assume u(ci) is weakly decreasing (from u(0) =

0), implying that the direct costs of consumption outweigh any direct benefits (more

on this soon) — thus, the net instantaneous return −1+u(ci) < 0 for all ci, so that it is

never desirable to face a decision. To optimize, the individual chooses the consumption

sequence (c0, c1, . . .) that maximizes lifetime utility:

U =∞∑i=0

δti(−1 + u(ci)), (1)

which is expressed relative to the outside opportunity (i.e. if the individual never faces

the decision, then U = 0), and where δ ∈ (0, 1) is a constant discount factor.

To complete the problem, we need to specify how the decision points depend on

past consumption levels. The interval function, denoted by τ , is thus defined as the

time-interval from the current decision point to the next:

τ(si, ci) = ti+1 − ti > 0,

where si ∈ [0, 1] is the habit stock — a proxy for past consumption that evolves

according to

si+1 = (1− σ)si + σci, (2)

for some σ ∈ (0, 1). Thus the habit stock at a given decision point is a weighted

average of the habit stock and the consumption level associated with the previous

decision point (observe ci = si maintains a constant habit stock).10 The “speed factor”

σ captures how quickly the habit changes with consumption. Without loss of scope10The form in (2) mirrors Laibson’s (2001) stock transition, in which “red” and “green” stock variables

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or insight, we will take σ > 0 to be suitably small, which reflects the notion that bad

habits (substance addictions, in particular) tend to develop gradually, requiring many

reinforcements over time to take full effect.11

We will now formalize cravings as decision points (i.e. a craving forces the individual

to consider consumption). Accordingly, letting a subscript denote the associated partial

derivative (e.g. τc = ∂τ(s,c)∂c

), properties of τ(s, c) reflect key properties of cravings

discussed in the introduction:

Assumption 1 τ(s, c) is twice-continuously differentiable and satisfies, for all s, c:

(i) τs(s, c) < 0,

(ii) τc(s, c) > 0,

(iii) s′ > s implies τ(s′, s′) < τ(s, s),

(iv) −δτ(s,c) is strictly concave with positive cross-partial derivative.

Part (i) says a larger habit stock entails more frequent decisions, ceteris paribus, which

captures the long-run reinforcement of past consumption. Part (ii) says, with s fixed,

the interval to the next decision increases with consumption, which captures the prop-

erty that consumption lessens cravings in the short-run;12 in light of the decision op-

portunity cost, τc > 0 embodies a short-term incentive to consume, as a means to delay

the next decision point. The below diagram illustrates the essence of parts (i) and (ii),

which are the key assumptions of the model:13

evolve only when a cue of their color is present. Thus the red stock does not decay when the green cueappears even though the red habit is not reinforced by consumption. The transition in (2) is analogouslylinked to the next decision, as opposed to a future time, so that a deliberate choice is needed for the habit toevolve. Used for mathematical cleanliness, omitting time from the transition is not essential for the results.

11The gradual nature of habituation in addiction is addressed by Robinson and Berridge (1993), Everittand Wolf (2002), and Kalivas and Volkow (2005). Appendix A.12 provides additional discussion and a formalelaboration on the small σ restriction, and shows that ruling out large σ does not limit the domains of τ andu that the model can accommodate. The small σ restriction also serves as a counterweight to the (upcoming)assumption that, while ti+1 − ti rises with ci, all intervals after ti+1 decrease with ci (holding all ci+1, ci+2 . . .constant). This lack of persistence in the near-term interval-extending effect of consumption was adopted forease of exposition and to obviate the need for multiple state variables.

12Houtsmuller and Stitzer (1999) show rapid smoking suppresses cravings. Dallery et al. (2003) find thelatency to smoke is substantially longer following rapid smoking relative to normal paced smoking. Zacny andStitzer (1985) similarly demonstrate the latency to smoke decreases after nicotine deprivation.

13Together, parts (i) and (ii) also embed a form of tolerance, i.e. the notion that as an addiction develops,more consumption is needed to get the same effect. This follows because τ(s′, c′) = τ(s, c) with s′ > s impliesc′ > c, which means that the stronger habit requires a higher consumption level to ward off the next cravingwith equal effectiveness.

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ti ti+1 ti+1

ti ti+1 ti+1

low c high c

low c high c

high s

low s

Part (iii) of Assumption 1 says if c = s (which holds in a steady-state), then si-

multaneously increasing both c and s by the same amount decreases the time between

decisions, i.e. the interval-shortening effect of s outweighs the lengthening effect of c

along the c = s frontier. Concavity of −δτ(s,c) from part (iv) implies diminishing re-

turns to consumption. To see why, first note −δτ(s,c) is the discounted value of the next

decision opportunity cost. Since the sole benefit of consumption is delaying this cost,

−δτ(s,c) can be regarded as the effective return function, and its concavity is analogous

to diminishing returns through a concave instantaneous utility function — a standard

assumption in discrete-time dynamic programming (e.g. Stokey et al., 1989). The

positive cross partial derivative of −δτ(s,c) implies that the effective marginal return

to consumption rises with habit strength, so that the extent to which consumption

alleviates cravings is greater for strong habits than for weak habits.

2.1 Static Consumption Preferences (or Lack Thereof)

We will now specify two cases of particular interest for the agent’s static consumption

preferences, as represented by u(c). First, as a simple benchmark for the basic cravings

model, we consider omitting the instantaneous utility function altogether:

Condition ZP [Zero Static Consumption Preferences] u(c) = 0 for all c.

This condition abstracts from static consumption preferences in that there are no

direct returns to consumption; instead, consumption matters solely through its effect

on the timing of future decisions. Thus ZP allows us to explore behavior driven entirely

by the interval function τ(s, c) along with the −1 decision opportunity cost.14

A limitation of ZP is that it does not account for the many direct costs of addictive

consumption (e.g. harmful health effects, social costs, and purchase price). To accom-

modate direct costs, let θ ≥ 0 denote a direct cost parameter and express instantaneous

utility as u(c|θ) with the normalization u(0|θ) = 0 for all θ. We presume that θ = 0

corresponds to ZP so that u(c|0) = 0 for all c. Now consider the alternative:

14To be sure, ZP is not considered for its realism, but to elucidate the role of endogenous decision pointsrelative to the role of preferences in understanding addiction.

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Condition NP [Negative Static Consumption Preferences] θ > 0, where u(c|θ) is

strictly concave and twice-continuously differentiable (in c and θ), with u′(c|θ > 0) < 0

and θ′ > θ implies u′(c|θ′) < u′(c|θ), for all c.

With NP, consumption is directly harmful in that the instantaneous utility function is

negative and decreasing.15 Throughout the paper, results for NP will generally hold as

long as consumption preferences aren’t “too negative,” i.e. for a range of small θ > 0.

Therefore, from here on, the phrase “under NP” is used as shorthand for “there is a

θ > 0 such that θ ∈ (0, θ) implies the following holds under NP.” This implicit qualifier

reflects the notion that if direct costs are too large, the individual will never consume

(and analyzing “behavior” in this scenario is an empty endeavor).

2.2 Dynamic Optimization

To emphasize the unique mechanics of endogenous decision points and for ease of

exposition, the dynamic optimization problem will be laid out for ZP. Although the

derivations are messier, it is straightforward to adapt what follows to NP.

Under ZP, dynamic optimization is characterized by the Bellman equation:

V (s) = −1 + maxc{δτ(s,c)V ((1−σ)s+ σc)}. (3)

Here V (s) is the value function at a decision point given habit stock s. At each decision

point, the individual pays the decision opportunity cost and chooses consumption to

maximize the present value of the next-decision value function. The decision oppor-

tunity cost is placed outside the maximization argument to emphasize its invariance

to chosen consumption (i.e. it is sunk). The choice of c determines the extent of dis-

counting through τ (where large c helps), and determines next decision’s habit stock

in the argument of the future value function (where large c hurts). The first-order and

envelope conditions are:

0 = σV ′(si+1) + ln(δ)τc(si, ci)V (si+1), (4)

and V ′(si) = δτ(si,ci)[(1−σ)V ′(si+1) + ln(δ)τs(si, ci)V (si+1)]. (5)

Let the superscript + denote the function associated with the next decision and15Since NP consolidates the direct returns from consumption to u(c), it is still an idealization in that

actual direct returns likely involve an immediate benefit and a larger, future cost. Because consolidation ofpresent and future returns with u′(c) < 0 means direct costs must outweigh consumption’s direct benefits, NPhandicaps the decision points model relative to standard addiction theories in two ways: (i) bad habits mustnow be motivated despite a “dislike” for consumption; and (ii) future costs cannot be diluted by sufficientlysteep impatience (low δ) to “rationalize” consumption. With constant discounting and time-separable utilitythis consolidation has no effect on optimizing behavior.

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++ to denote the next-after-next decision (e.g. τ+ = τ(si+1, ci+1), τ++ = τ(si+2, ci+2),

etc.). Given s0, if an interior optimal consumption path exists, it is defined by the

Euler equation:16

στ+s − (1− σ)τ+

c

στ+s − (1− σ)τ+

c + τc= 1 + δτ

+

[στ++

s − (1− σ)τ++c

στ++s − (1− σ)τ++

c + τ+c

]. (6)

2.3 Steady-States

To characterize potential steady-states, define the movement function (for ZP) as:

µ(s) =

[τc +

σ(τs + τc)δτ

1− δτ

]c=s

, (7)

where the arguments of τ(s, c), τs(s, c), and τc(s, c) are suppressed. The first term in

brackets, τc, is proportional to the marginal benefit of increasing consumption from c =

s by marginally delaying the next decision point; the second term, σ(τs+τc)δτ/(1−δτ ),

is proportional — by the same factor as the first term — to the marginal cost of

increasing consumption from c = s, given a policy of maintaining a constant habit

stock at all subsequent decision points.17 Thus, the movement function µ(s) gives the

net marginal return to consumption at c = s, provided the individual consumes exactly

the next-decision habit stock thereafter. Since the optimization problem is Markovian,

if c = s is optimal now, then the habit stock permanently remains at s and the policy

for future consumption suggested by µ(s) is indeed optimal. It follows that if s is an

interior steady-state under ZP, then µ(s) = 0. If µ(s) > 0, the habit stock rises to

the next decision because it would then be optimal to consume more than s, while

the habit stock falls if µ(s) < 0 by equivalent logic. Hence, the sign of µ(s) indicates

the direction the habit stock moves from the current decision to the next under ZP;

equivalently, letting c(s) denote the optimal consumption level given s, the signs of

µ(s) and of (c(s)− s) will agree.

Assumption 2 µ(s) is continuously-differentiable and satisfies:

(i) µ(0) ≤ 0.

(ii) µ(s) > 0 for some s ∈ (0, 1).

(iii) µ(1) < 0.

16The derivation of the Euler equation is provided in Appendix A.1.17The interpretation of the first term is evident from the partial derivative of the effective payoff function,−δτ , with respect to c: − ln(δ)δττc. For the second term, note that σ(τs + τc) is the marginal decreasein the interval between all future decision points and −δτ/(1 − δτ ) is the future value term given that thenext-decision habit stock is maintained thereafter.

10

Part (i) implies that abstinence is optimal at the minimum habit stock; if the inequality

binds, then abstinence happens to be exactly optimal under ZP in that c(0) = 0

independent of the fact that zero is the lower bound of c’s domain. From part (ii),

there is at least one value of s from which the habit will optimally grow (i.e. c(s) > s).

From part (iii) the habit will optimally weaken from the maximum habit stock (i.e.

c(1) < 1); put differently, the strongest possible habit is too severe to be rationally

sustained as a long-run outcome.18

Assumptions 1 and 2 are maintained throughout and all results will hold for both

ZP and NP, except where otherwise noted.19 All proofs are in the Appendix.

Proposition 1 [Addicted Steady-State] There is a unique s∗ ∈ (0, 1) along with a

s∗L ∈ [0, s∗) such that si converges to s∗ for all s0 ∈ (s∗L, s∗), and to zero for all s0 < s∗L.

Proposition 1 establishes a stable steady-state of addiction, s∗ > 0. The individual

becomes addicted in the long-run if s0 exceeds a “tipping point” s∗L, which is an unstable

steady-state characterized by low (possibly zero) consumption. Together, these steady-

states reflect the “slippery slope” nature of addiction in that one can readily fall into

its trap, but escape is difficult. Behavior of an addict is defined jointly by the steady-

state consumption level c∗ = s∗ and the intervals between consumption occasions, τ ∗ =

τ(s∗, c∗), which fits with pharmacological evidence that addiction involves consistent

doses separated by consistent time-intervals (Benowitz, 1991). Addiction is bad here

because it entails frequent, distracting “thoughts” of consumption (i.e. payments of

the decision opportunity cost).20

The next result characterizes two measures of habit strength:

Proposition 2 [Measures of Habit Strength]

(i) c(s) is increasing in s, for all s.

(ii) τ ∗ < τ(s, c(s)), for all s < s∗.

From part (i) of Proposition 2, the per-decision consumption level increases with habit

strength. Meanwhile, part (ii) asserts that consumption is more frequent under addic-

18Each part of Assumption 2 can be motivated in light of basic real-world observations: part (i) fits withthe fact that most people do not smoke, part (ii) reflects the fact that many people do still end up on the pathto addiction, and part (iii) suggests that even the most nicotine-dependent cigarette addicts could feasiblysmoke more than they actually do.

19A parametric example that satisfies all assumptions and accompanying discussion are provided in theappendix of Landry (2012).

20The notion that addiction can be bad simply by being a “waste of time” is a unique feature of the modelunder ZP. That said, while its mechanism is novel, this specification captures a harmful addiction in thetraditional sense of Stigler and Becker (1977) and Becker and Murphy (1988), in that a higher stock of pastconsumption implies lower future utility, i.e. V ′(s) < 0. Under NP, however, the model motivates addictionunder a stronger concept of “harmful” than previous treatments in that consumption is directly costly too,i.e. u′(c) < 0 and V ′(s) < 0.

11

tion than for weaker habits. This joint level-and-frequency characterization of habit

strength fits with evidence that frequent cigarette smokers inhale more nicotine per

cigarette than infrequent smokers (Shiffman, 1989; Shiffman et al., 1994), and are also

substantially more likely to chain-smoke multiple cigarettes in succession.21 As men-

tioned in the introduction, standard habit-formation and addiction models, which are

cast in standard, metronomic time, account for the level but not the frequency aspect

of habit strength. Notions outside economics likewise capture one out of two aspects,

as the standard measure in other behavioral disciplines is based on frequency only

(Ouellette and Wood, 1998; Verplanken, 2006).22

2.4 Comparing Chippers and Addicts

To illustrate some key themes, we draw on the notion of a chipper — a label commonly

applied to occasional smokers. Here a “chipper” will broadly refer to a user who is in

the vicinity of s∗L > 0, but for simplicity will formally connote someone exactly at s∗L,

with steady-state consumption levels denoted by c∗L = s∗L and associated intervals by

τ ∗L = τ(s∗L, s∗L). The contrast in chippers’ and addicts’ consumption patterns highlights

the joint level-and-frequency distinction between strong and weak habits:

. . .

. . . addict

chipper

τ ∗

τ ∗L

c∗L

c∗

c

t

Here, each dot here represents a single instance of consumption. In line with previously-

cited evidence, the lower height of chippers’ dots indicates a lower consumption level per

21The latter type of evidence — which suggests that addicts average more cigarettes per occasion thanchippers — is codified in DSM-V, the American Psychiatry Association’s official diagnostic manual, as it listschain-smoking as a criterion of nicotine dependence. Chase and Hogarth (2011) in fact regard chain-smokingas one of the two DSM-V symptoms that most accurately reflect habit-formation. Also see Colby et al. (2000),who note that chain-smoking “may best exemplify” the DSM-V criterion that pertains to time spent usingdrugs.

22Still, these studies suggest that the frequency measure is an incomplete psychological representation sincebehavior often becomes automatized as habits develop, which raises interesting modeling questions. Namely,the automatization of choices reduces (but does not eliminate) their toll on decision-making faculties, suggest-ing there is gray area in classifying a “decision.” (If a “choice” is performed automatically without deliberation,is it preceded by a decision?) Automatization could be accommodated in the current setup by letting thedecision opportunity cost decrease with s (given c > 0), yet its relevance for explaining observable consump-tion patterns is unclear. As Baker et al. (2004) argue, automatization may not be particularly significantfor understanding the motivations of addiction and that, more often than not, addicts are largely aware of“deciding” to use drugs during use.

12

occasion, and the larger distance between neighboring dots indicates chippers consume

less frequently than addicts.

The next figure shows how optimal consumption may look with and without direct

costs (for the case in which s∗L = 0 under ZP):

c(s), ZP

c(s), NP

s

c

s∗L s∗ s∗

As seen above (for NP), the chipper steady-state signifies a rough tipping point for ad-

diction — an idea proposed by medical researchers (Benowitz and Henningfield, 1994).

The instability of the chipper steady-state also fits with evidence that, relative to occa-

sional smokers, the positions of nonsmokers and of frequent smokers are comparatively

stable (Zhu et al., 2003).23 In comparing the two preference conditions, the diagram

also shows how integrating direct costs can bolster the relative standing of chippers,

which fits with empirical trends as growth in the direct costs of tobacco consumption

has coincided with the rise of chippers to nearly 40% of U.S. smokers.24,25

23Recall, chippers do not exist exactly at s∗L, but instead in its vicinity, where the chipper used for formalillustrations represents an occasional user who sits perfectly on the fence between quitting and becomingaddicted. Although s∗L may not attract a significant population share in the long-run, someone who startsclose to (but not at) s∗L may remain nearby for a long time because habits move slowly when close to asteady-state. Also, since many chippers are only “approximate” chippers, the likelihood a smoker is classifiedas such would decrease with age, in line with evidence that the chipper population is disproportionately young(Hennrikus et al. 1996; Wortley et al., 2003).

24It is clear that chippers are more prominent with direct costs in the above figure because chippers don’texist under ZP for the special (illustrative) case in which µ(0) = 0. While not as obvious, the interpretationremains valid for µ(0) < 0, in which case chippers would exist under ZP. Even in this case, s∗L would increasewith θ from θ = 0, making it harder to justify a bimodal characterization that implicitly lumps chippers withabstainers at a steady-state in the vicinity of zero. Furthermore, the range of s0’s such that s converges to zeroin the long-run widens as θ increases; hence, we would expect the prevalence of users to fall, and consequentlythe prominence of chippers to rise relative to that of addicts.

25Since the 1960s, several costs and restrictions have proliferated — e.g. new taxes, better informationregarding health risks, the emergence of a social stigma, and smoking bans at work or in public areas (Beckeret al., 1994; Chaloupka and Warner, 2000). See Shiffman (2009) for evidence on chippers. The 40 percentfigure is based on a recent Health and Human Services survey (Substance Abuse and Mental Health ServicesAdministration, 2010). While the survey does not use the term “chipper,” the statistic holds for the usualdefinition as nondaily smokers.

13

2.5 Curiosity’s Role in Initiation

Why would a nonuser ever start consuming? Following the standard line of reasoning

from rational addiction theory (Becker and Murphy, 1988), we could attribute initia-

tion to factors that increase the marginal return to consumption. However, we cannot

simply reiterate the factors that Becker and Murphy mention — e.g. divorce, unem-

ployment and death of a loved one — to account for initiation in the current setting

because the relevant return in rational addiction theory is a stock-dependent utility

function, while here it is our effective return function, −δτ(s,c), that gives the discounted

next-decision opportunity cost. Moreover, before entertaining factors that may increase

the marginal return through −δτ(s,c), we first need an adequate explanation for why,

at very low s, this return would be increasing in c to begin with. That is, we cited

cravings to justify τc > 0 (which is necessary and sufficient for −∂δτ(s,c)/∂c > 0), but

for someone who has never consumed, cravings are an unsatisfying justification.

A plausible reason for why τc > 0 would hold at low s is curiosity. Like a craving,

curiosity is ‘appetitive’ (Loewenstein, 1994); in particular, abstinence is unlikely to

satiate a curiosity to try the good, making it difficult to forget about in the short-run,

yet consumption would conversely alleviate a nagging curiosity. Curiosity can therefore

influence behavior in qualitatively the same manner as cravings. Thus, by equating an

intense curiosity to a relatively large magnitude for τc > 0 at small s, the model can

motivate initiation.26 Moreover, this basis for initiation fits with related findings. For

instance, Milton et al. (2008) report that curiosity is the most frequently-cited motive

to try cigarettes among those who try, while Pierce et al. (2005) conclude from survey

data that curiosity may be a “critical precursor to smoking.” Furthermore, Pierce et

al. find that having friends who smoke and receptiveness to tobacco advertising are

positively associated with curiosity. Thus, by piquing a curiosity, peers or advertise-

ments (or both) could increase the return to consumption via −δτ(s,c) at low s, in turn

increasing the prevalence of new smokers.

3 Interval-Driven Adjacent Substitution

An important and well-established feature of addictive consumption is “adjacent sub-

stitution,” which means that demand in the near-term future decreases with current

26Formally, we could let idiosyncratic curiosity directly affect the habit stock so that a fleeting circumstancecan get someone “hooked,” similar to how Becker and Murphy (1988) consider letting stressful events (suchas divorce, unemployment, death of a loved one) directly affect the habit stock.

14

consumption — i.e. the good now and in the “adjacent” future are substitutes.27 Adja-

cent substitution is evident, for instance, from Rose et al.’s (1984) finding that smokers

prefer higher nicotine intake following two hours of nicotine deprivation. Bickel et al.

(1998) similarly show that price increases trigger “drug seeking” behavior, in which

an addict continues to crave a good until it is consumed — i.e. addicts substitute to

future (and presumably cheaper) consumption — and Erskine et al. (2010) report that

smokers attempting to abstain for a week increase consumption during the following

week, relative to a control group.

To formally define adjacent substitution in a standard, discrete-time setting, the

relevant demand measure is consumption in the next decision period. However, next

decision’s consumption level is an inconsistent standard in the current framework be-

cause the intervals between decisions vary endogenously. Since demand ought to reflect

how much and how often one consumes, a time-averaged consumption rate is a natural

measure. Decision-i demand is thus defined as

x(i) =ci

τ(si−1, ci−1),

which is the mean consumption rate during the union of decision i’s ‘active’ and ‘latent’

periods; i.e. the full demand period includes the decision point and the waiting period

that immediately preceded it, (ti−1, ti].28Adjacent substitution holds if next-decision

demand falls with current consumption: −∂x(i+1)∂ci

> 0, where the left-side of the in-

equality is the marginal rate of adjacent substitution (MRAS), which quantifies the

degree of substitutability between the good now and the good in the immediate future.

Adjacent complementarity holds if MRAS is negative (i.e. the good now and in the

immediate future are complements). Since the level and the frequency of consumption

at the next decision point both enter decision-i+ 1 demand, at the outset MRAS’ sign

may be attributable to the effect of ci on either ci+1 or on τ(si, ci) (or both).

Proposition 3 [Interval-Driven Adjacent Substitution] Adjacent substitution holds at

all s, provided c(s) > 0.

Proposition 3 establishes interval-driven adjacent substitution in that the interval-

27A second key demand pattern, “distant complementarity,” is addressed in Section 4. The present emphasison adjacent substitution is based on: (i) its connection to the incentives that drive addiction; (ii) its novel,embedded concept of commitment; and (iii) its omission in prevailing theories. Each of these reasons will soonbecome clearer.

28Put differently, decision-i demand is decomposed as the active demand at the decision point ti and thelatent demand during the latency period ending at ti. The term “latent demand” is borrowed from Mor-genstern’s (1948) description of interim demand as “latent” for a consumer who must wait until a particularfuture time to purchase goods.

15

extending effect of consumption is the basis for adjacent substitution; this particular

form of adjacent substitution matches previously-cited laboratory evidence that the

latency period to the next instance of consumption rises with present consumption.

Furthermore, through its interval-driven feature, adjacent substitution is in essence a

manifestation of the incentive for addictive consumption. Observe MRAS is increasing

in the magnitude of τc > 0, which in turn proxies for the strength of this incentive; thus,

a greater degree of adjacent substitution reflects a more powerful short-term incentive

to consume, ceteris paribus.

The interval-driven aspect of adjacent substitution also embeds an unusual depic-

tion of commitment : by delaying the next opportunity to consume, consumption is a

valuable short-term commitment strategy to restrict future consumption.29 Because

avoiding a decision point allows the individual to engage outside opportunities, com-

mitment and flexibility arrive in tandem — a contrast to the usual treatment in which

they are opposites, characterized by constricted or expanded choice sets. This decision-

level characterization of commitment reflects the notion that considering the primary

decision generally precludes (at that moment) full engagement in outside opportuni-

ties, such as considering a different consumption decision. As it pertains to cravings,

decision-level commitment arises because a neglected craving is an ongoing drain on

decision-making faculties, and is inherent in any approach that postpones or eliminates

cravings — such as wearing a nicotine patch to facilitate quitting.30

Even without its interval-driven feature, adjacent substitution still bucks conven-

tion, as adjacent complementarity is the definition of addiction in Becker and Murphy’s

(1988) standard rational addiction theory based on habit-formation preferences. The

standard tool in microeconomics for modeling habits, habit-formation preferences are

represented by a time-inseparable utility function in which past consumption enters

today’s return, e.g. u(c, s) with uc > 0 and usc > 0 presumed.31 Adjacent com-

29Here, consumption is analogous to the canonical representation of illiquidity as a commitment device inLaibson’s (1997) hyperbolic discounting model, in which, upon the sale of an asset, illiquidity postpones theopportunity to consume from its proceeds.

30Since it reduces an addict’s frequency of cravings to that of a less addicted individual, a nicotine patchcould be modeled as a (beneficial) device that lowers the “effective” habit stock by some φ > 0. In thiscase, the interval function would become τ(s − φ, c), so that the decision points and consumption choices ofa patch-wearing individual would be equivalent to a patchless scenario with a lower habit stock (abstractingfrom potential boundary issues with τ undefined for s < 0).

31Habit-formation preferences are the prototype of time-inseparability (history-dependence). As Pollak(1970) describes his seminal habit hypothesis: “(i) that past consumption influences current preferences andhence, current demand and (ii) that a higher level of past consumption of a good implies, ceteris paribus, ahigher level of present consumption.” Time-inseparable preferences continue to be central in the extensivehabit-formation literature that followed — including variations on Becker and Murphy’s rational addictiontheory (Orphanides and Zervos, 1995; Orphanides and Zervos, 1998; Gruber and Koszegi, 2001; Laibson, 2001)as well as recent treatments not focused on addiction, such as Rozen’s (2010) axiomatization and Crawford’s

16

plementarity is manifest in habit-formation preferences because the marginal utility

of consumption rises with past consumption. Although many alternatives to ratio-

nal addiction have been proposed, there have been no serious objections to adjacent

complementarity.

4 Measurement and Inference in Discrete-time

This section considers a simple discrete-time empirical analysis of behavior from en-

dogenous decision points, allowing us to compare the model to prevailing treatments

of bad habits in a common temporal framework. As choices are typically aggregated

into metronomic bins in empirical work, the exercise will also help to distinguish what

traditional discrete-time inference can and cannot tell us when the timing of decisions

varies endogenously.

4.1 Detecting Adjacent Substitution

Consider a discrete-time setup (presuming the decision points model is the data gener-

ating process), where ` is the fixed measurement period length and Tn is the end point

of measurement period n. Then measured demand for measurement period n, denoted

as Cn, is its average demand level:32

Cn =1

`

∫ Tn

Tn−`x(i : ti−1 < t ≤ ti) · dt.

Accordingly, measured adjacent substitution is defined in a typical discrete-time man-

ner: −∂Cn+1

∂Cn> 0, which implies substitutability between consecutive measurement

periods, i.e. the measured MRAS is positive.33 Per usual, measured adjacent comple-

mentarity holds if the inequality is reversed.

Proposition 4 [Measured Substitution Threshold]

There is a `0 > 0 such that demand exhibits measured adjacent substitution if ` < `0

and measured adjacent complementarity if ` > `0, at all s such that c(s) > 0.

(2010) revealed-preference empirical treatment of time-inseparability.32Implicit in this formulation is that the flow demand is constant over each decision’s demand period. This

“flattening” allows a continuous measure of C. If the relevant instant associated with decision-i demand isthe time of the purchase for consumption at ti, the specification can be motivated by assuming the purchaseis equally likely to occur at any time in (ti−1, ti].

33As is, measured MRAS’ denominator is not well defined because there are multiple ways to perturb demandin n if the measurement period contains multiple decisions. Qualitatively, it will not matter for the resultswhether the perturbed demand is at the beginning of the measurement period, at the end, or at some fixedfraction in between. For simplicity and without loss of generality, perturbed consumption is assumed to be atthe end of the measurement period, so that measured MRAS can be expressed as − ∂Cn+1

∂x(i)where ti = Tn.

17

That is, if the measurement period length exceeds a threshold, behavior falsely appears

to satisfy adjacent complementarity — a reflection of actual distant complementarity.

Below the threshold, adjacent substitution is detected. Thus, the result shows how,

with sufficiently high-resolution data, the decision points model is distinguishable from

rational addiction theory and habit-formation preferences even in discrete time. This

qualitative dependence of the nature of intertemporal substitution on data resolution

is shown below:

measuredMRAS

meas. periodlength (`)

`0

Rational Addiction/Habit-Form. Preferences

Endog. Decision Points

The substitution threshold `0 from Proposition 4 may serve as a crude measure for

the duration of abstinence-induced withdrawal because such withdrawal is character-

ized by an escalation of cravings, which encourage consumption. Addictive substances

are characterized by relatively long periods of withdrawal — three to four weeks for

nicotine and up to ten weeks for cocaine (Hughes et al., 1994). Long withdrawal peri-

ods fit with (but do not directly reveal) the strong incentive inherent in the underlying

interval-driven adjacent substitution because there is less reason to consume if with-

drawal is short-lived. The length of nicotine withdrawal is also compatible with Erskine

et al.’s (2010) evidence that substitution persists at least one week into the future, as

smokers consume more following a week of attempted abstinence relative to a control

group.

The substitution threshold may also have policy relevance in that `0 represents the

minimum duration of a temporary consumption ban such that the ban does not lead

to an increase in demand immediately after it is in effect, relative to demand in the

absence of a ban. Thus, if a policy-maker’s objective is to reduce consumption, bans

longer than `0 will provide continued benefits after they expire, while bans shorter than

`0 are undermined by post-ban increases in consumption. This latter prediction is not

shared by the standard model based on adjacent complementarity, which predicts that

18

any temporary ban would cause post-ban demand to drop, regardless of its duration.

Consistent with the decision points model, Evans et al. (1999) find that daily con-

sumption falls by less than the share of the time during the workday when a workplace

smoking ban is instituted, implying that consumption increases outside working hours.

While a helpful start, measured adjacent substitution masks the richer description

of interval-driven adjacent substitution. Further, because the endogenous intervals

(and decision opportunity costs) are undetected, discrete-time measurement conceals

the incentive to consume — regardless of dataset resolution. This is not a trivial

omission considering theoretical interest in addiction largely stems from the challenge

of explaining why a rational agent would consume harmful addictive products. As the

agent’s motivation is invariably hidden in a metronomic world, the next subsection

examines how, in order to rationalize behavior, preferences are molded as they are

inferred in discrete-time.

4.2 Biases in Discrete-Time Inference

Empirical analyses generally seek to understand observed behavior in terms of eco-

nomic primitives. Hence, as a natural next step, preferences are inferred using the

misspecified measurement framework. Given that the incentive remains concealed,

the inferences may shed light on how harmful addictions are rationalized in metro-

nomic models. The procedure will “reveal” both static consumption preferences and

time preferences, i.e. discounting. For reasons that will soon be clear, we highlight

two departures from constant discounting that are especially prominent in research

on addiction: (i) commodity-dependence, as addicts appear exceedingly impatient to

consume drugs, discounting future drugs much steeper than future money;34 and (ii)

endogenous time-preference, as drug use appears to improve short-run patience and

lessen patience in the long-run — this latter effect is evident in theories of endoge-

nous time preference in which addictive consumption steepens impatience (Becker and

34For instance, Bickel et al. (1999) find that smokers discount cigarettes more than money, while Maddenet al. (2007) and Kirby et al. (1999) similarly find that heroin addicts discount heroin more than money.Khwaja et al. (2007) provide evidence of commodity dependence even within the health domain, reportingthat smokers are as patient as nonsmokers regarding their willingness to undergo a colonoscopy.

19

Mulligan, 1997; Orphanides and Zervos, 1998).35,36

As a conventional approach to infer preferences, consider a contemporaneous payoff

function U+(C) and a decreasing future loss function U−(C) where C is demand in the

current measurement period. For ease of exposition (and without loss of generality),

the future loss occurs one time-unit after the payoff. Thus the composite return is:

U(C|δ) = U+(C) + δU−(C).

This form reflects the usual contention that addictive consumption entails present

benefits and future costs (beyond the indirect costs of habit reinforcement). Now

suppose, with the true discount factor δ, the composite return accurately represents

time-aggregated static consumption preferences.37 Then U(Cn|δ) is any positive affine

transformation of ∫ Tn

Tn−`

[δtiu(ci)

ti − ti−1

: ti−1 < t ≤ ti

]dt.

That is, with the correct δ, the composite return over time-aggregated demand is

equivalent to the true time-aggregated direct returns from demand in the measurement

period. Based on the known payoff and future loss functions, the measured discount

factor for period-n, denoted by δn, is calculated from the empirical utility maximization

problem:

δn = {δn : Cn = arg maxC

U(C|δn)}.

Thus the measured discount factor is the discount factor for which observed demand

appears to maximize utility. In turn, the measured marginal utility of consumption can

35The short-term effect is documented for smokers by Field et al. (2006) and for heroin addicts by Giordanoet al. (2002). Conversely, evidence that ex-users exhibit substantially more patience than current users isprovided by Bickel et al. (1999) for cigarette users, and by Bretteville-Jensen (1999) for users of heroin andamphetamines. While noting the possibility that selection may account for the divergence, Bretteville-Jensenwrites: “great impatience and short-sightedness will probably also arise as a result of addiction.” Bickel etal. likewise conclude: “never- and ex-smokers could discount similarly because cigarette smoking is associatedwith reversible increasing in discounting or due to selection bias.”

36A third and well-known constant discounting departure, present bias will not be formally addressed here.Typically motivated by the demand for commitment, as in Gruber and Koszegi’s (2001) integration of quasi-hyperbolic discounting into rational addiction theory, present bias is not covered for two reasons. First,commitment value has already been motivated from endogenous decision points without time-inconsistency.Second, as in Gruber and Koszegi’s framework, constant discounting and quasi-hyperbolic discounting areindistinguishable in the current estimation procedure. That said, Section 6 motivates an empirical test thatdifferentiates the original rational addiction model, present-biased addiction models, as well as the decisionpoints model.

37As used here, static consumption preferences means the choice is considered in isolation from future choicesand thus reflects only the direct costs of consumption. The results derived from this specification are robustto inclusion of a future value term (along with U−(C)) to account for indirect costs. A future value term isomitted for simplicity and also to adhere to standard practice, as measured discount rates typically do notaccount for habit reinforcement (Frederick et al., 2002).

20

be computed at any demand level from

U ′(C|δn) = U (+)′(C|δn) + δnU(−)′(C|δn),

which characterizes the shape of the individual’s inferred consumption preferences un-

der time-aggregation.

The next result shows how endogenous decision points can reconcile apparent con-

stant discounting departures, while shedding light on the inference of consumption

preferences when “time” is misspecified.

Proposition 5 Under NP, assume U+ and U− are both weakly concave.

(i-a) [Impatience for Good] δn < δ, provided Cn > 0.

(i-b) [‘False Positive’ Preferences] U ′(C|δn) > 0 for all C ∈ [0, Cn].

(ii-a) [Endogenous Discounting] δn+1 falls with Cn if ` > `0, and rises with Cn if ` < `0,

where `0 is the substitution threshold in Proposition 4.

(ii-b) [Time-Inseparability] For any C ≥ 0: U ′(C|δn+1) increases with Cn if ` > `0,

and decreases with Cn if ` < `0.

Proposition 5 highlights four measurement biases from discrete-time inference, which

arise because estimation misattributes decision-point influences to the discount and

utility functions. In particular, part (i-a) shows how consumption is justified by ex-

cessive impatience for the good, which reconciles commodity-dependent discounting

since the estimate is less than the true δ (which is presumably measurable in other

domains). Consequently, positive consumption preferences are inferred in part (i-b),

as future costs are seemingly neglected. Part (ii-a) captures both long- and short-run

forms of endogenous discounting. Estimated patience falls with “yesterday’s” demand

if the period length is too large to detect adjacent substitution (otherwise, estimated pa-

tience rises). Because the discount factor enters the composite return, time-inseparable

consumption preferences follow in part (ii-b).38

While matching regularities from empirical research, the inferences from Proposi-

tion 5 also encompass the default theoretical tools to explain bad habits in standard

time. To start, the impatience-based rationalization of positive consumption prefer-

ences in part (i) — a reversal of the underlying NP condition — resembles a key

ingredient of rational addiction theory, which employs steep discounting to dilute the

future harmful effects of consumption.39 The long-run form of time-inseparability in

38Apparent time-inseparability can also arise if, instead of taking U+ as known and estimating δ, any δ > 0is presumed (whether correct or incorrect) and the form of U+ is inferred.

39As Becker and Murphy (1988) explain: “this paper relies on a weak concept of rationality that does not

21

part (ii-b) captures the second key ingredient: habit-formation preferences. This de-

piction of habit-formation preferences as an artifact has neuropsychological backing, as

researchers find drugs are increasingly ‘wanted’ despite not being increasingly ‘liked’

as addiction develops.40 The long-run form of endogenous discounting in (ii-a) reflects

the history-dependent discount functions used by Orphanides and Zervos (1998) and

by Becker and Mulligan (1997).

Since discrete-time inferences from endogenous decision points resemble fundamen-

tally different preference structures, the exercise suggests that metronomic time may

have empirical limitations. While real time or time use data may be a useful start, it

would help if decision points could be detected. (A method to elicit decision points is

proposed in Section 5.)

4.3 Implications for Welfare

As Proposition 5 helps convey, the decision points model has implications for empiri-

cal welfare analysis. As underscored by the inference of positive consumption prefer-

ences under NP, the current treatment suggests that preferences inferred from time-

aggregated demand are not aligned with true preferences and thus may not be a valid

indicator of static well-being.41 This complication arises because the analyst can’t “see”

the underlying costs and benefits of consumption in discrete-time. Although it poses

new data burdens for empirical welfare calculations, the model retains the classical link

between choice and well-being, alleviating a common welfare concern in the addiction

literature.42 Namely, the model has an unambiguous welfare standard (lifetime utility,

as given in (1)), which is not generally the case when using the aforementioned default

modeling tools. For instance, commodity-dependent discounting violates unitary time-

preference, which precludes an objective standard for aggregating utilities over time.43

rule out strong discounts of future events.” This reliance can be viewed as a consequence of standard time’sexogenous decision schedule, as metronomic decision-making imposes a bound of sorts on rationality in thatit grants agents no control over the timing of their own decisions.

40In economic parlance, ‘liking’ naturally maps to the true consumption preference and ‘wanting’ to demand.The liking-wanting gap is a centerpiece of the incentive-sensitization addiction theory of Robinson and Berridge(1993). Also see Robinson and Berridge (1998, 2008), Berridge and Robinson (2000), and Bernheim and Rangel(2004) for additional background and discussion.

41While they attribute this misalignment to mistakes, Bernheim and Rangel (2004) also argue that prefer-ences revealed from behavior can systematically diverge from true preferences.

42In an externality-free environment, there would be no role for taxation or other restrictions on consumption.However, if decision points arise due to phenomena other than cravings, policies that deter consumption canimprove consumer welfare. For instance, if advertisements induce unwanted decision points, an advertisingban will benefit consumers. If peer consumption induces a decision point, consumption bans may similarlybenefit consumers, especially if peer groups are large and heterogeneous with respect to habit strength — seeLandry (2013) for details.

43That is, should utilities be aggregated according to the discount rate for cigarettes or the discount ratefor money? Although commodity-dependence is not explicitly a feature of rational addiction theory, it is not

22

Questioning the reliability of welfare assessments from endogenous preferences, as in

his canonical habit-formation preferences model (Pollak, 1970), Pollak (1978) argues

that the fundamental theorem of welfare economics — “that in competitive equilibrium

everyone gets what he wants” — loses its normative significance because it may then

be a mere corollary of the basic tenet that “people come to want what they get.”

5 External Cues

Exposure to an item or context associated with a behavior can act as a powerful impetus

to do the behavior. For addiction, the significance of such external cues, such as an

offer of a cigarette or the sight of an ashtray, is widely documented in psychology — see

Caggiula et al. (2001) or Carpenter et al. (2009). This line of research has motivated

economic cues models by Laibson (2001) and Bernheim and Rangel (2004). While the

representation of a cue varies from model to model, stochasticness is a shared feature

because cues introduce variance in consumption patterns (and long-term outcomes),

as use is linked to seemingly random environmental factors.

Although it can be difficult to differentiate a cue from a craving in isolation, a clear

contrast emerges when comparing their roles across the habit spectrum.44 That is,

unpredictable factors (i.e. external cues) are germane at low levels of dependence, but

less so in the realm of heavy addiction. A trademark of strong nicotine dependence

is inflexible smoking patterns largely uninfluenced by environmental cues (Shiffman et

al., 2004). Accordingly, consumption is more strongly associated with external cues

for chippers than for addicts (Shiffman and Paty, 2006); this pattern “significantly

distinguishes” addicts from chippers (Shiffman and Sayette, 2005).

The diminished influence of external cues for stronger habits is also evident in the

disproportionate effect of advertisements on younger, newer smokers, as Pollay et al.

(1996) estimate teenagers’ sensitivity to cigarette advertisements is roughly triple that

of adults. Furthermore, Cummings et al. (1997) report that over 90 percent of ado-

lescent smokers (versus 35-40 percent of adults) smoke one of the top-three selling

brands — Marlboro, Camel, and Newport, which are also the most heavily advertised

— while generic brands (which use minimal advertising) capture a five-fold greater

share of adult than adolescent demand.

unrelated either in light of estimated discount rates from empirical rational addiction analyses — 56 to 223percent in Becker et al. (1994) — that dwarf plausible values, as implied by market rates of return on financialinvestments.

44Since external cues can elicit internal cravings, to avoid confusion, I use “craving” to refer to an urge withan internal origin and “cue” to refer to an urge with an external origin.

23

5.1 Cues as Decision Points

The influence of a single cue is reminiscent of an internal craving: when a cue arises,

the urge to consume can suddenly materialize. In light of the similarities, external cues

are modeled as (stochastic) decision points. As a simple baseline, assume cues have

a fixed arrival rate λ > 0.45 The deterministic element (cravings) is retained between

the original model and the “natural” interval function τ(s, c), which is the time until

the next decision if no cues arise in the interim.

The next lemma establishes an equivalence from the original, deterministic model

to the new setup that combines stochastic cues with internal cravings.

Lemma 1 Consider the stochastic cues model described above with arrival rate λ and

natural interval function τ . Define

τ 0(s, c) =ln[(λ− e−λτ(s,c) ln(δ)δτ(s,c)

)/(λ−ln(δ))

]ln(δ)

.

Then, given s0, the optimal consumption sequence (c0, c1, . . .) with stochastic cues is

the same as in the deterministic setting with the interval function τ 0.

Lemma 1 defines τ 0 as a “certainty equivalent” for the random interval function τ , in

that the optimization problem with stochastic cues is the same as in the cravings-only

model using τ 0. That is, at a decision point, the expected future value with cue-arrival

rate λ and natural interval τ(s, c) equals the known future value with τ 0(s, c), for all

s, c. The lemma allows us to carry over previously-derived steady-states to the cues

setting.46 Hence steady-state terminology (“addicts” and “chippers”) is maintained

with the understanding that the corresponding habit stocks are those that would exist

in the original model using τ 0.

5.2 Diminished Role of Random Cues under Addiction

The next result compares the importance of stochastic cues between addicts and chip-

pers:

45Two notes on fixed λ: first, the fixed arrival rate treats the duration of the interruption as negligible.Second, there are reasons to expect the true cue-arrival rate varies with s and c, e.g. smokers are likelyexposed to smoking situations more often than nonsmokers. The results of the section are robust to anendogenous cue-arrival rate, provided its curvature is weak relative to the interval function’s.

46As before, si+1 =(1−σ)si+σci regardless of whether ti+1 is craving- or cue-induced (without this simplifyingassumption, the deterministic equivalence is lost). In the new cues setting, Assumptions 1 and 2 are maintainedfor τ . Increasing λ from zero will have (qualitatively) the same effect on steady-states as increasing θ fromzero, so that the existence of a chipper steady-state no longer requires Assumption NP.

24

Proposition 6 The following are smaller for addicts than for chippers, ceteris paribus:

(i) the variance of τ .

(ii) the elasticity of demand with respect to λ (or, equivalently, the share of consumption

that coincides with an external cue).

Proposition 6 captures the observation that addicts have regimented patterns of drug

use, while the influence of cues is more pronounced for occasional users, who have

less predictable consumption routines.47 In particular, part (i) indicates that addicts’

consumption schedules are less random than chippers’; and part (ii) establishes that

addicts’ demand is less associated with stochastic cues, which holds because stronger

habits are more dependent on (i.e. involve a higher frequency of) internal cravings.

Interpreted in terms of advertisements, part (ii) suggests that the elasticity of demand

with respect to advertising frequency would be greater for those with less-developed

habits than for those with well-established habits, which again fits with previously-cited

evidence. Both parts of Proposition 6 are robust to a degree of habit stock dependence

in the cue-arrival rate — e.g. the addict may have a smaller share of consumption

coinciding with a cue even if the addict’s arrival rate exceeds the chipper’s.48

The current treatment contrasts recent cue-based economic theories’ emphasis on

random external factors as the primary drivers of addiction. In Laibson’s (2001) and

Bernheim and Rangel’s (2004) models, the path to addiction is marked by an escalating

vulnerability to stochastic environmental cues. This runs counter to the evidence that

consumption becomes predominantly driven by internal cravings as addiction develops,

as the intervals between doses become shorter and more predictable.

5.3 Cues as Deterrents to Consumption

The next result shows that, for low-frequency cues, increasing the cue-arrival rate λ

has polar effects on the two observable measures of habit strength:

Proposition 7 There is a λ′ > 0 such that, increasing λ with λ ∈ [0, λ′] has the

following effects (for both addicts and chippers):

(i) the expected consumption frequency rises,

(ii) the per-decision consumption level falls.

47The variance of the consumption frequency, Var[τ−1] is not defined, but it would also be negativelycorrelated with s in the continuous limit under a discretization of time units that precludes τ = 0.

48We can use the formula in Lemma 1 to define τ0 from an endogenous cue-arrival rate, λ(s, c), and τ0 willstill be a function of s and c only. Lemma 1’s proof does not require fixed λ, hence the “certainty equivalent”interpretation of τ0 remains valid and analysis can proceed as it would with fixed λ.

25

From part (i), the consumption frequency rises with λ, which holds mainly because

more cues implies more decision points. However, consumption levels shrink from part

(ii), which follows from the fact that, by competing with cravings, cues reduce the

incentive to consume. For example, increasing consumption from c to some c′ > c

elongates the natural interval to τ(s, c′) > τ(s, c), but a cue’s arrival prior to τ(s, c)

negates this benefit of higher consumption.

When isolating the response to a single cue, part (ii) of Proposition 7 implies a

“novelty effect” in that cue-induced consumption levels are higher if cues are rare than

if they are common. While the hypothesis has not been directly tested, Niaura et al.

(1992) provide evidence that supports the basic prediction that “more” cues can reduce

the propensity to consume, as exposure to an olfactory smoking cue reduces the urge

to smoke when a visual cue is already present.49 The result is also in line with evidence

of “wearout” in advertising, whereby consumers can become annoyed and unresponsive

to advertisements that are repeated too frequently.50

This general notion that rapidly-arriving cues can deter consumption can be taken

a step further:

Proposition 8 Fix θ > 0 given NP. There is a λ > 0 such that λ > λ implies c(s) = 0

for all s.

Overexposure to cues can fully desensitize the individual in that consumption levels

fall to zero for sufficiently large λ. Abstinence becomes optimal because, extending

the logic from part (ii) of Proposition 7, sufficiently rapid cues can fully wipe out the

incentive to consume by ensuring that the next decision will be cue-induced.51 Since

overexposure to cues means consumption will cease, the result further deemphasizes

the importance of cues for understanding addictive behavior.

It is worth noting that Proposition 8 pertains specifically to exogenous cues, as

abstinence is justified by the individual’s helplessness against the barrage of cues.

While some cues in reality may come and go regardless of the individual’s choice, the

next subsection describes how a special type of cue can feature a strong dependence

49In this study, olfactory and visual cues refer, respectively, to the smell and sight of someone smoking. Alaboratory subject would have less incentive to smoke when both cues are present if seeing precedes smellingor if the smell of smoke lingers beyond initial exposure because, when the visual cue arrives, the presence ofthe olfactory cue means it is unlikely that the next decision point will be craving-induced.

50See, for instance, Cacioppo and Petty (1979), Calder and Sternthal (1980), and Pechmann and Stewart(1989). In contrast, standard information- or preference-based advertising models do not account for wearoutin advertising.

51Cues can also reduce and even eliminate (as λ → ∞) the indirect disincentive to consume due to τs < 0.Since choices only matter through their effect on future cravings under ZP, consumption choices cease to affectutility as λ→∞, so that any c ∈ [0, 1] is a solution. This indeterminacy is why Proposition 8 only applies toNP (in which case the direct disincentive due to u′(c) < 0 is preserved even as λ→∞).

26

on past choices, which gives rise to starkly different implications than exogenous cues

for behavior and for long-run outcomes.

5.4 The Good as a Cue

The motivational force of a cue is commonly attributed to the notion that cues signal

the availability of the good.52 Therefore, the good itself ought to be a cue. Presuming

unconsumed quantities do not perish, a modification is needed to represent the good

as a cue because unlike the good, cues come and go regardless of chosen consumption

in the existing setup. To incorporate its persistence, suppose a(t) is the amount of

the good that is immediately accessible at t and λa > λ is the cue-arrival rate at all

t with a(t) > 0, where λ now denotes the arrival rate when a(t) = 0. If the amount

of the good at some decision point is positive and slightly greater than the “usual”

optimal consumption level (without the good as a cue), then it is optimal to consume

all of the good: ci = a(ti) > c(si), where c is maintained as the optimal consumption

function without cue-persistence. Access to the good creates an imperative to raise

consumption because the unconsumed good invites a heightened arrival rate, from λ

to λa.53

Because its persistence motivates higher than usual consumption, access to the

good can precipitate undesirable long-run outcomes. For instance, addiction becomes

inevitable for an otherwise ambivalent chipper who consumes a > s∗L, or even for a

“recovering” user on the path to abstention with s < s∗L, provided (1 − σ)s + σa >

s∗L. These observations fit with evidence that the physical presence of the good often

drives smoking and alcohol relapses (Niaura et al., 1988). Furthermore, the incentive

to consume no longer relies on the “soft persistence” of internal cravings, which is

consistent with the finding that cigarette cues undercut the effectiveness of quitting

aids that inhibit cravings, such as the nicotine patch.54 Because immediate access to

large quantities is especially burdensome, the good as a decision point also offers a

rationale for why, despite a price premium, smokers commonly buy packs of cigarettes

instead of cartons (Khwaja et al., 2007). With the good as a persistent cue, restricting

52The importance of availability in cue effectiveness is addressed by Droungas et al. (1995), Juliano andBrandon (1998), and Carter and Tiffany (2001).

53The concept parallels Fudenberg and Levine’s (2012) consideration of persistent temptations. In theirmodel, an agent suffers a continual cost from the availability of a persistently tempting good (until it is gone).Consequently, it is “easier to resist” a fleeting temptation than a persistent temptation.

54In Waters et al. (2004), abstinent smokers randomized to a nicotine patch or placebo were given a cigaretteand instructed to hold it as if they were between puffs. The patch lowered baseline urge to smoke, but it didnot attenuate the spike in the urge from holding a cigarette. As the authors write, “abstinence-related urgesand cue-provoked urges may have different origins and might be treated differently.”

27

access can be a viable commitment strategy to reduce the decision-point frequency.55

6 Availability Schedules and Decision-Point Elicitation

Since access to the good can induce extra, unwanted decision points, yet utility-

maximizing consumption levels can be positive, we cannot make a blanket statement

regarding whether availability is good or bad. To formally clarify the circumstances

for which availability helps more than it hurts, now suppose that the good is available

at t if and only if there is a cue at t. This formulation can be related to a setting in

which the good is available only through another individual (more on this soon). The

availability schedule will refer to the set of times for which the good is available, and

will be taken as given ex-ante with respect to consumption decisions in that chosen

consumption levels may endogenously depend on the availability schedule, but not vice

versa.

Proposition 9 The utility-maximizing availability schedule coincides exactly with the

equilibrium decision point sequence from the basic, cravings-only model.

To consider how this utility-maximizing availability schedule might be elicited, sup-

pose a subject (presumably a smoker) can only access cigarettes through offers from

an experimenter for the duration of a sufficiently long experiment. Also suppose that

when cigarettes are offered, the subject may smoke or decline, but hoarding is not

allowed. If the subject can choose the offer times up front, in effect choosing the avail-

ability schedule, Proposition 9 implies that the chosen such offer times would be the

subject’s (natural) future decision points. Thus, even if decision points are not directly

observable, the result points to the possibility that they can be experimentally elicited.

Such decision-point elicitation could be useful in light of the inherent limitations in

measurement and in inference with standard, time-aggregated data (see Section 4).

In addition to eliciting a user’s natural decision points, the offer-availability paradigm

described above (and simple variations of it) could be used to measure or test other

key properties of the model:

• Variation in decision point frequencies between strong and weak habits could be

measured by comparing chippers’ and addicts’ chosen availability schedules. If

55This example also illustrates how the two main requisites for consumption — the availability of the goodand a decision point — can be intertwined. As mentioned in Section 3, since a decision point is a requisite forconsumption, commitment is inherent in any method to prevent decision points. It follows that cue avoidancerepresents a valuable form of commitment, where the associated value is largely commensurate to the cue’stendency to persist in the face of abstinence.

28

addicts opt for more frequent offers, the outcome would appear to support the

model’s first key assumption: τs < 0 (Assumption 1, part (i)).56

• If a period of abstinence is imposed at the beginning of the experiment, comparing

the chosen availability schedule after abstinence to that of a control could reveal

how decision point frequencies depend on recent consumption levels. If imposed

abstinence increases subjects’ ensuing chosen offer frequencies, ceteris paribus, the

outcome would appear to support the second key assumption: τc > 0 (Assumption

1, part (ii)).57

• The decision opportunity cost could be estimated from nonusers’ valuations of

different availability schedules. In particular, a nonusers’ willingness-to-accept for

an availability schedule with n + 1 offers in lieu of a better availability schedule

with n offers would in principle measure the (discounted) decision opportunity

cost.58

• To quantify the incentive to consume as a means to decrease the short-run decision

point frequency, the estimated decision opportunity cost could then be combined

with the short-run increase (relative to the control) in the volume of smokers’

elicited decision points immediately following imposed abstinence. To be precise,

this quantity would reflect the dis incentive to abstain from the (unconstrained)

consumption levels among those in the control group, during the pre-trial period

in which abstinence is imposed on the treatment group.

• To estimate the cumulative decision opportunity costs — i.e. the total distraction

costs — associated with having an addiction, we could multiply the estimated

decision opportunity cost by addicts’ inferred decision point frequencies in the

initial procedure.

Beyond its implications for measurement, Proposition 9 also suggests a new test to

distinguish the decision points model from prevailing theories of addiction. That is, the56Individual habituation could allow the analyst to quantify the link between decision point frequencies and

distant past consumption levels, but would likely be too slow to reliably detect in a single session. However,measuring multiple subjects’ baseline levels of cotinine — a biomarker for nicotine — offers a potential meansto link decision-point frequencies to quantifiable measures of habit strength. Recently, economists have startedto use cotinine data to enrich conventional data sources in empirical analyses of cigarette addiction (e.g. Addaand Cornaglia, 2006, 2010). Also see Gine et al.’s (2010) field study, in which cotinine data is used to inferbehavior.

57Comparing consumption patterns (i.e. levels and latencies) between the two conditions could also serveas an empirical test of interval-driven adjacent substitution (Proposition 3).

58Here it would be particularly important to ensure that subjects can engage in activities of similar valueto that of outside opportunities (relative to smoking) in their natural environments. One possible approachwould be to recruit subjects who spend a substantial portion of their day-to-day lives on the internet, andprovide internet access during the experiment.

29

result implies that a user would desire periodic future access to the good, separated

by temporary periods without access. (If the user is in a steady-state, the desired

availability schedule would be evenly spread across time.) In contrast, availability is

always desirable in standard rational addiction theory (Becker and Murphy, 1988),

so a user would prefer offers as often as possible.59 In Gruber and Koszegi’s (2001)

version with present-biased discounting, however, a time-inconsistent user would not

want access at any time beyond the present.60 These contrasts (illustrated below) can

be interpreted in terms of the temporal pattern of precommitment. Namely, a user

would never desire precommitment in the original rational addiction theory, while the

long-term “self” would prefer permanent precommitment in the present-biased model.

The decision points approach instead emphasizes the distinction in whether or not

the agent anticipates considering the decision as the determinant in whether or not

precommitment is desirable at a given future time.

nowt

Rational Addiction

Decision Points

Present Bias

Desired Availability Schedules

7 Conclusions

This paper departed from the standard modeling practice that takes as given the

frequency with which an agent considers a particular consumption decision. In doing so,

a few ideas were proposed to explain where such “decision points” may come from in the

realm of addiction, and how choices can influence the timing of future decision points.

The framework provides an alternative to standard time’s inherent “metronomism,”59Access is always preferable in the original rational addiction theory, which is cast in continuous-time.

Based on a liberal interpretation of a discrete-time version, along with an assumption that consumptionoccurs at the same relative moment within every period, one could argue that rational addiction theory couldbe compatible with the periodic access prediction. That said, rational addiction would still not fit with manyaspects of Proposition 9: (i) the decision points model only establishes fixed intervals between access in asteady-state, while this interpretation of rational addiction theory would require fixed intervals even for userswho are not at a steady-state; (ii) outside the times access is strictly preferred, rational addiction would implyindifference between access and no access, precluding a positive willingness-to-pay for precommitment at alltimes, while no access is strictly preferable at such times in the decision points model; (iii) only the decisionpoints model would permit individual variation in desired offer frequencies, while rational addiction wouldpredict that chippers and addicts, for example, would desire the same availability schedule.

60Future availability is similarly undesirable when it is linked to cues in Laibson’s (2001) and Bernheim andRangel’s (2004) models.

30

which, as argued, is an overlooked imposition that limits the range and explanatory

power of economic models.

From the general notion of a “cue” — exposure to an item or context associated with

a behavior — other decision point models can be readily motivated.61 For instance, as

alluded to in the text, advertisements can be modeled as decision points. Advertise-

ments and other promotional strategies may be particularly important in explaining

where some individuals’ first decision points come from. Consistent with the tobacco

industry’s youth-targeting strategies, advertising to a child who has never actively con-

sidered smoking (even if aware that cigarettes exist) could rouse a profitable new habit,

simply by inducing the initial decision point.62 The use of marketing-induced decision

points to recruit new customers similarly fits with DiFranza and McAfee’s (1992) ob-

servation that the tobacco industry “repeats the word ‘decision’ like a mantra.” Even

industry-sponsored “prevention” programs, which consistently draw attention to the

decision to smoke, appear to be a sly method of propagating decision points among

susceptible youth.63 As an example, the Tobacco Institute’s first educational program,

Helping Youth Decide, purports to help “young teenagers learn to make more of their

own decisions.”

A follow-up to this paper, Landry (2013), considers endogenous decision points in a

multi-agent setting in which peer consumption induces a decision point. The group

model captures three commonly-observed aspects of conformity: (i) herd-behavior

within groups, e.g. one individual’s idiosyncratic choice to try an addictive good can

lead her peers to likewise consume for the first time; (ii) self-sorting into homogeneous

groups, driven by the desire of infrequent users to avoid frequent users due to the fre-

quent, unwanted decision points they inflict; and (iii) synchronization of consumption,

as exemplified by social smokers who predominantly smoke in unison with others. This

group decision point model also motivates insistent peer pressure and shows how, by re-

lenting only when its target agrees to consume, peer pressure incentivizes consumption

in a similar manner as cravings.

The decision point appears to be a versatile modeling device, capable of unifying

61Besides focusing on specific types of cues, the cues-as-decision points concept is amenable to theoreticalexplorations in non-addiction contexts. Landry (2014) proposes a unified theory for the endowment effectand present bias, based on the premise that situational cues — in particular, those that subjects typicallyencounter in laboratory settings — induce decision points.

62Marketers of Camel cigarettes have proven particularly adept at making inroads with the very young, asover 90 percent of six-year olds knew the smoking cartoon “Joe Camel” (Camel’s ex-mascot) — a recognitionlevel on par with Mickey Mouse — in a study by Fischer et al. (1991).

63Landman et al. (2002) classify the portrayal of smoking as an “adult decision” as the overarching theme ofyouth-directed industry programs. The authors also report that Philip Morris (sellers of Marlboro’s) conductedover 130 youth “prevention” campaigns in over 70 countries, as of 2001. Unsurprisingly, Farrelly et al. (2002)find intentions to smoke rise after exposure to a Philip Morris youth anti-smoking campaign.

31

a wide range of phenomena that are important in the realm of bad habits. Surveying

the existing literatures in economics on habits, addiction, advertising, and group peer

effects (also known as “social interactions”), a consistent pattern is revealed. Namely,

for all of these applications, the prevailing approaches fall into one of two categories:

(i) models that are based on endogenous (or otherwise nonstandard) preferences, and

(ii) models that work through information or beliefs. While its usefulness in contexts

besides “bad habits” remains largely unexplored, the decision point thus represents

an alternate tool that may offer a new understanding of behaviors that are currently

captured through preference- or information-based models.

A Appendix

A.1 Derivation of the Euler Equation

The Euler equation (6) is derived from the Bellman equation (3), the first-order con-

dition (4), and the envelope condition (5) as follows: First, substitute out V ′+ in (5)

using (4). Second, in the new expression for (5), iterate every term forward one deci-

sion. Again, substitute out V ′+ (after the iteration) using (4). Third, substitute out

V ++ using (3) once-iterated — i.e. using the Bellman with the left-side set to V +.

After solving for V +, insert the expression and its once-iterated variant for V ++ into

the left- and right-sides, respectively, of (3) to get the Euler equation (6). Note, the

left-side of the Euler equation is the absolute value of the next-decision value function

−V + > 0.

A.2 Proof of Proposition 1

It will help to work with the following change-of-variables, substituting out consump-

tion, and instead allowing consecutive habit stock variables enter as the arguments of

the interval function:

τ(si, si+1) = τ(si, ci) = τ(si,

si+1−(1−σ)siσ

)To denote the partial derivatives of the newly-expressed interval function, the subscripts

1 and 2 are used for the current and the next-decision habit stocks, respectively:

τ1(si, si+1) =∂τ(si, si+1)

∂si= τs(si, ci)−

(1− σ)τc(si, ci)

σ

τ2(si, si+1) =∂τ(si, si+1)

∂si+1

=τc(si, ci)

σ.

32

Expressing the problem in terms of the habit stock sequence, the Bellman (under ZP)

is

V (si) = −1 + maxsi+1

{δτ(si,si+1)V (si+1)},

where si+1 ∈ [(1 − σ)si, (1 − σ)si + σ]. The corresponding first-order and envelope

conditions are, in order:

0 = V ′(si+1) + ln(δ)τ2(si, si+1)V (si+1)

V ′(si) = ln(δ)τ1(si, si+1)δτ(si,si+1)V (si+1)

Plugging the once-iterated envelope condition into the first-order condition (to elim-

inate V ′(si+1)), then substituting out V (si+2) using the once-iterated Bellman, and

rearranging gives

V (si+1) =−τ1(si+1, si+2)

τ1(si+1, si+2) + τ2(si, si+1).

Plugging the above expression and its once-iterated version into the once-iterated Bell-

man gives the Euler equation:

τ1(si+1, si+2)

τ1(si+1, si+2) + τ2(si, si+1)= 1 +

δτ(si+1,si+2)τ1(si+2, si+3)

τ1(si+2, si+3) + τ2(si+1, si+2)(8)

By undoing the change-of-variables so that the above is expressed in terms of τ , τs,

and τc instead of τ , τ1, τ2, etc., it is readily verifiable that equation (8) is equivalent to

the original Euler equation (6).

Let {s∗i } = (s∗0, s∗1, . . .) denote the sequence that satisfies equation (8) given s∗0 = s0,

and let {si}∞i=0 = (s0, s1, . . .) be any other feasible sequence of habit stocks (in which

si+1 ∈ [(1 − σ)si, (1 − σ)si + σ] for all i = 0, 1, . . .) given s0. Now let D define the

difference between the lifetime utilities with {s∗i } and with {si}:

D =I∑i=0

δti − δt∗i

To show that the sequence that satisfies the Euler equation, {s∗i }, is optimal, it suffices

to show that D ≥ 0. By concavity of −δτ , and with s∗0 − s0 = 0, we have

33

D ≥ limI→∞

I∑i=1

[i∑

k=0

−∂δt∗i

∂s∗k· (s∗k − sk)

]

= limI→∞

I∑i=1

ln(δ)δt∗i

[τ1(s∗i , s

∗i+1)(s∗i−si)−

i∑k=1

(τ1(s∗k, s∗k+1) + τ2(s∗k−1, s

∗k))(s

∗k−sk)

]Factoring out − ln(δ) and collecting each s∗i − si term, we get

D ∝ limI→∞

I∑i=1

δt∗i

[τ1(s∗i , s

∗i+1)δτ(s∗i ,s

∗i+1)

(I∑

j=i+1

δt∗j−t∗i+1

)+ τ2(s∗i−1, s

∗i )

(I∑j=i

δt∗j−t∗i

)](s∗i−si)

= limI→∞

I∑i=1

δt∗i

[τ1(s∗i , s

∗i+1)

(−1+

I∑j=i

δt∗j−t∗i

)+τ2(s∗i−1, s

∗i )

I∑j=i

δt∗j−t∗i

](s∗i−si).

Now define Xi =∑∞

j=I+1 δt∗j−t∗i , so that limI→∞ Xi = 0 for all i. Then we can use:

∞∑j=i

δt∗j−t∗i =

τ1(s∗i , s∗i+1)

τ1(s∗i , s∗i+1) + τ2(s∗i−1, s

∗i ),

which gives:

D ≥ limI→∞

I∑i=1

δt∗i

[τ1(s∗i , s

∗i+1)

(−

τ2(s∗i−1, s∗i )

τ1(s∗i , s∗i+1) + τ2(s∗i−1, s

∗i )−Xi

)+ τ2(s∗i−1, s

∗i )

(τ1(s∗i , s

∗i+1)

τ1(s∗i , s∗i+1) + τ2(s∗i−1, s

∗i )−Xi

)](s∗i − si)

= limI→∞

{−

I∑i=1

δt∗i[τ1(s∗i , s

∗i+1) + τ2(s∗i−1, s

∗i )]Xi(s

∗i − si)

}= 0

Hence the Euler equation (6) is sufficient for the optimal consumption sequence.

Thus, the steady-state condition derived from the Euler equation, µ(s) = 0, is sufficient

for an interior steady-state. It follows that µ(0) ≤ 0 with c ∈ [0, 1] guarantees s = 0

is a steady-state. From continuity of µ and parts (ii) and (iii) of Assumption 2, an

interior steady-state is guaranteed since µ(s) = 0 must hold for some s ∈ (0, 1) because

µ is positive for some s in this range and negative at s = 1. Thus, s∗ = min{s ∈ (0, 1) :

µ(s) = 0 and µ(s − ε) > 0 for sufficiently small ε > 0}. Since µ(s) crosses zero from

above, s∗ is locally stable.

To establish the above for NP, we can use the NP Bellman, given by V (s) = −1 +

34

maxc {u(c|θ)+δτ(s,c)V ((1−σ)s+σc)}, to derive the first-order and envelope conditions:

0 = u′(ci|θ) + δτ(si,ci)[σV ′(si+1) + ln(δ)τc(si, ci)V (si+1)],

and V ′(si) = δτ(si,ci)[(1−σ)V ′(si+1) + ln(δ)τs(si, ci)V (si+1)].

The Bellman, first-order, and envelope conditions for NP can be combined as before to

obtain a NP Euler equation for which its sufficiency for the optimal consumption path

follows as in the ZP case. Setting c = s throughout, we can derive the steady-state

condition. Define

µθ(s) = µ(s)− u′(s|θ)[1− (1−σ)δτ(s,s)]

ln(δ)δτ(s,s)(1− u(s|θ))< µ(s).

It is readily verifiable that µθ(s) is the NP movement function analog so that µθ(s) =

0 implies s is a steady-state under NP. We also have that µθ(0) < µ(0) ≤ 0 and

µθ(1) < µ(1) < 0. Further, limθ→0+ µθ(s) = µ(s), which implies (with continuous

differentiability of u with respect to θ) that µ(s) > 0 implies µθ(s) > 0 for sufficiently

small θ. Thus, parts (i)-(iii) of Assumption 2 for the ZP movement function ensures

the same properties hold for the NP analog. In turn, using the same logic as before,

Proposition 1’s characterization of steady-states holds as in the ZP case. �

A.3 Proof of Proposition 2

A.3.1 Proof of Proposition 2, part (i)

For c(s) to be increasing in s for all s, we need the Bellman’s maximization argument,

u(c) + δτ(s,c)V ((1 − σ)s + σc), to be supermodular, i.e. satisfying the property of

increasing differences. That is, for any c′ > c, if u(c′) + δτ(s,c′)V ((1−σ)s+σc′)− u(c)−δτ(s,c)V ((1−σ)s+σc) is increasing in s, Topkis’ Theorem (Topkis, 1978) establishes the

desired monotonicity in c(s). Hence, we need:

(1−σ)δτ(s,c′)V ′[(1−σ)s+ σc′] + ∂∂s

[δτ(s,c′)]V [(1−σ)s+ σc′]

− (1−σ)δτ(s,c)V ′[(1−σ)s+ σc]− ∂∂s

[δτ(s,c)]V [(1−σ)s+ σc] > 0.(9)

Note since u(·) is independent of s the condition is the same for ZP and NP. Now define

∆V (y, e) = V (y + e) − V (y) and ∆V ′(y, e) = V ′(y + e) − V ′(y). Then rearranging

terms in (9) gives:

(1−σ)(δτ(s,c′)−δτ(s,c)

)V ′[(1−σ)s+σc′]+

(∂∂s

[δτ(s,c′)−δτ(s,c)])V [(1−σ)s+σc′]

+ ∂δτ(s,c)

∂s

[(1−σ)∆V ′[(1−σ)s+σc, σ(c′−c)]+∆V [(1−σ)s+σc, σ(c′−c)]

]> 0

(10)

35

as the condition for supermodularity. The limit of the left-side of (10) as σ tends to

zero is (δτ(s,c′)−δτ(s,c))V ′(s)+ ∂∂s

[δτ(s,c′)−δτ(s,c)]V (s), where ∆V and ∆V ′ both converged

to zero with σ.

We know τc > 0, which implies δτ(s,c′)−δτ(s,c) < 0. Therefore (δτ(s,c′)−δτ(s,c))V ′(s) >

0. We know ∂2

∂s∂c[−δτ ] > 0, which implies ∂

∂s[δτ(s,c′)− δτ(s,c)] < 0. Therefore ∂

∂s[δτ(s,c′)−

δτ(s,c)]V (s) > 0. Since both terms are positive, supermodularity holds and from Topkis’

Theorem, c(s) must be increasing at all s for sufficiently small σ. �

A.3.2 Proof of Proposition 2, part (ii)

Take s ∈ (0, s∗). Since µ(s∗) = 0, µ(s) > 0. It follows that c(s) > s, which implies

τ(s, s) < τ(s, c(s)) because τc > 0. Since τ(s∗, s∗) < τ(s, s) given s∗ > s, this gives

τ(s, c(s)) > τ(s∗, s∗). �

A.4 Proof of Proposition 3

MRAS is proportional to −στ(si, c(si)) + c(si+1)τc(si, c(si)).64 The small σ restriction

ensures the second term, c(si+1)τc(si, c(si)) > 0, dominates so that MRAS> 0. �

A.5 Proof of Proposition 4

Without loss of generality, suppose n = i = 0, T0 = t0 = 0, and s0 = s∗. Let

I = max i : ti < `, i.e. tI is the last decision point in the interior of the measured

demand period. First consider ZP. Then

C1 =1

`

∫ `

0

x(i : ti−1 < t ≤ ti)dt =1

`

(I∑i=1

ci

)+

(`− tI)cI+1

`τ(sI , cI).

Now measured adjacent substitution (complementarity) holds if the measured MRAS

is positive (negative). Since only sign matters, these conditions are equivalent if the

measured MRAS is multiplied by the constant ` > 0 (replacing mean with total de-

mand in the numerator). That is, measured adjacent substitution holds if −∂[`·C1]∂c0

> 0.

For any ` < τ ∗, measured adjacent substitution is satisfied because −∂[`·C1]∂c0

> 0 iff

−∂[`·x(1)τ∗]∂c0

> 0 iff −∂x(1)∂c0

> 0, which holds by assumption.

64As in this proof, comparative statics are expressed using (partial) derivatives for clarity. However, allproofs can be worked out without drawing on differentiability in the same way except replacing any partialderivative ∂f(c, ·)/∂c with (f(c′, ·)−f(c, ·))/(c′−c) for c′ sufficiently close to c (where the signs are the same ifc′ > c and reversed if c′ < c), and c′(s) by (c(s′)− c(s))/(s′− s), where s′ is the habit stock associated with c′.

In this proof, for instance, the condition for adjacent substitution would become c((1−σ)s+σc(s))τ(s,c(s))

≷ c((1−σ)s+σc)τ(s,c)

for c ≷ c(s).

36

Since ∂ck∂c0

= c′(s∗)∂sk∂c0

and ∂s1∂c0

= σ, it follows that:

∂sk∂c0

= σ[(1−σ) + σc′(s∗)]k−1,∂ck∂c0

= σc′(s∗)[(1−σ) + σc′(s∗)]k−1.

With ∂t1∂c0

= τ ∗c , we have ∂tk+1

∂c0= ∂tk

∂c0+ ∂τ(sk,ck)

∂c0= ∂tk

∂c0+ τ ∗s

∂sk∂c0

+ τ ∗c∂ck∂c0

, which, for all

k ≥ 2, can alternatively be expressed as

∂tk∂c0

= τ ∗c + (τ ∗s + c′(s∗)τ ∗c )1− [(1− σ) + σc′(s∗)]k−1

1− c′(s∗).

Now [(1 − σ) + σc′(s∗)] ∈ (0, 1) follows from Proposition 2 part (i), along with

Assumption 2, part (ii)’s implication that c(s∗ + ε) < s∗ + ε for ε > 0. Therefore, 0 <∂sk+1

∂c0< ∂sk

∂c0and 0 < ∂ck+1

∂c0< ∂ck

∂c0< 0, for all k ≥ 1. Since ∂τ(sk,ck)

∂c0= σ(τ ∗s+c′(s∗)τ ∗c )[(1−

σ)+σc′(s∗)]k−1 with τ ∗s +c′(s∗)τ ∗c < τ ∗s +τ ∗c < 0, we have 0 > ∂τ(sk+1,ck+1)

∂c0> ∂τ(sk,ck)

∂c0, for

all k ≥ 1. Hence, ∂tk∂c0

< 0 implies ∂tk+1

∂c0< 0.

Now suppose there is no k > 1 such that∂tk∂c0

< 0. Then ∂tk∂c0

> 0 for all k > 1.

Since lifetime utility under ZP is U = −∑δti, we have ∂U

∂c0= − ln(δ)

∑∂ti∂c0δti > 0,

which contradicts the optimality of c(s∗) = s∗. Thus, there must be a k > 1 such that∂tk∂c0

< 0. Now,

∂[`C1]

∂c0

=I∑i=1

∂ci∂c0

+`− tIτ 2

(τ∂cI+1

∂c0

− cI+1∂τ(sI , cI)

∂c0

)− ∂tI∂c0

· cI+1

τ.

If ∂tI∂c0

< 0, then every term is positive. Therefore, −∂[`C1]∂c0

> 0 for all ` > tk, so

that measured adjacent complementarity holds for sufficiently large `. Now observe

−∂2[`C1]∂`∂c0

= − 1τ2

(τ ∂cI+1

∂c0− cI+1

∂τ(sI ,cI)∂c0

), which is finite and strictly negative for I > 0.

Hence, since −∂[`C1]∂c0

> 0 for small `, and −∂[`C1]∂c0

< 0 for large `, and −∂[`C1]∂c0

is strictly

decreasing in `, there exists a `0 such that −∂[`C1]∂c0

> 0 for all ` < `0 and −∂[`C1]∂c0

< 0

for all ` > `0. The result extends to NP due to continuity of all terms in the small θ

limit. �

A.6 Proof of Proposition 5

A.6.1 Proof of Proposition 5, part (i-a)

Since Cn maximizes U(Cn|δn), U(Cn|δn) ≥ U(0|δn). By NP, U(0|δ) > U(Cn|δ). Adding

the inequalities, substituting U(C) = U+(C) + δU−(C), and rearranging gives (δ −δn)(U−(Cn)− U−(0)) < 0. Since U− is decreasing, δ < δ. �

37

A.6.2 Proof of Proposition 5, part (i-b)

By the first-order condition that defines δn, concavity of U , and U(Cn|δn) > U(C|δn)

for all C 6= Cn, U(C|δn) is strictly increasing on [0, Cn]. �

A.6.3 Proof of Proposition 5, part (ii-a)

Solving for δn+1 after taking the first-order condition of the utility maximization prob-

lem for which it is defined gives δn+1 = −U (+)′(Cn+1)/U (−)′(Cn+1). Here, by concavity

of both, U (−)′ < 0 implies U (+)′ > 0 (otherwise Cn+1 > 0 would not satisfy measured

optimality). Since u is strictly concave, U is too, which means either U+ or U− is

strictly concave (or both are). Therefore, δn+1 must fall with Cn+1 since the magni-

tude of its numerator decreases and that of its denominator increases (and at least one

of these monotonicities is strict). Since −E[∂Cn+1

∂Cn

]≷ 0 for ` ≶ `0 (Proposition 4),

increasing chosen consumption at T (n) decreases δn+1 if ` < `0 and increases δn+1 if

` > `0. �

A.6.4 Proof of Proposition 5, part (ii-b)

Since ∂U ′(C|δ)/∂δ = ∂[U (+)′(C)+δU (−)′(C)]/∂δ = U (−)′(C) < 0, U ′(C|δn+1) moves in

the opposite direction as δn+1. Hence, the desired result follows from (ii-a). �

A.7 Proof of Lemma 1

Let τD denote any deterministic interval function. We first show there exists a τD such

that the optimization problems with τD and with stochastic τ are equivalent in that

c(s) is the same in both cases. A crucial element of this equivalence is that the stock

transition, si+1 = (1−σ)si + σci, is retained in the stochastic cues model. Therefore,

given si and ci, si+1, does not depend on whether ti+1 is craving-induced at ti + τ or

cue-induced before then. Hence the optimization problem at ti+1 is also independent

of whether ti+1 is craving- or cue-induced, since si+1 is the sole state variable.

Now dynamic optimization with stochastic cues follows:

V (s) = −1 + maxc{u(c) + E[δτ(s,c)]V ((1−σ)s+ σc)}.

The next-decision value function is factored out of the expectation because its value

depends only on s and c. Comparing this to the deterministic Bellman

V (s) = −1 + maxc{u(c) + δτ

D(s,c)V ((1−σ)s+ σc)},

38

we see the optimization problems are equivalent if and only if δτD

= E[δτ ]. Thus, we

need to show δτ0

= E[δτ ], where τ 0 ≡ ln[(λ− e−λτ ln(δ)δτ )/(λ−ln(δ))]/ ln(δ). With the

known distribution of τ , the expectation of δτ is computed by decomposing it into its

cue- and craving-components:

E[δτ ] =

∫ τ

0

δtλe−λtdt+ Pr(τ = τ) · δτ︸ ︷︷ ︸e−λτ ·δτ

=λ− e−λτ ln(δ)δτ

λ− ln(δ).

Thus, as seen from the definition of τ 0, we have δτ0

= E[δτ ]. �

A.8 Proof of Proposition 6

A.8.1 Proof of Proposition 6, part (i)

Assimilating our steady-state notation with the new notation, note τ ∗L > τ ∗, i.e. the

chipper’s natural interval is greater than the addict’s. Now compute:

Var[τ ] =

∫ τ

0

t2λe−λtdt− τ 2e−λτ −[∫ τ

0

tλe−λtdt+ τ e−λτ]2

=1−e−2λτ−2λτe−λτ

λ2.

Variance is higher at τ ∗L than at τ ∗ because ∂Var[τ ]/∂τ = 2e−2λτ (e−λτ + λτ − 1)/λ is

positive for all τ > 0. �

A.8.2 Proof of Proposition 6, part (ii)

Since consumption levels do not depend on the “type” of decision point, the share

of consumption that is cue-induced is the probability that a decision point will be

cue-induced, 1− e−λτ, which is higher for τ = τ ∗L than for τ = τ ∗. �

A.9 Proof of Proposition 7

Part (ii) is proven first because the result will be used in the proof of part (i).

A.9.1 Proof of Proposition 7, part (ii)

First note limθ→0+∂µθ(s)∂λ

= ∂µ(s)∂λ

since limθ→0+ u′(c|θ) = 0. Now let s ∈ {s∗L, s∗} be

either nonzero steady-state. Then

∂µ(s)

∂λ∝ ∂

∂λ

[τ 0c + δτ

0

(στ 0s − (1− σ)τ 0

c )],

39

where all functions are evaluated at s and c = s. Using τ 0 = ln[(λ−e−λτ ln(δ)δτ )/(λ−ln(δ))]ln(δ)

,

we get that, for x ∈ {s, c}, τ 0x = e−λτ δτ τx(λ−ln(δ))2

λ−e−λτ ln(δ)δτ, which implies

∂µ(s)

∂λ∝ ∂

∂λ

[τc(λ− ln(δ)) + (λ− e−λτ ln(δ)δτ )(στs − (1− σ)τc)

]∝(λ− ln(δ))[τc + (1 + τ e−λτ ln(δ)δτ )(στs − (1− σ)τc)].

Using the fact that limθ→0+ µθ(s) = 0 at a steady-state, we get

∂µ(s)

∂λ∝(στs − (1− σ)τc)[− ln(δ) + e−λτ ln(δ)δτ + τ e−λτ ln(δ)δτ (λ− ln(δ))]

∝(1 + τ(λ− ln(δ)))e−λτδτ − 1.

Define Z(δ) = (1 − τ ln(δ))δτ . Therefore at λ = 0 and given s is a steady-state,∂µ(s)∂λ

< 0 if and only if Z(δ) < 1. Note that Z(1) = 1. Since δ ∈ (0, 1), we havedZ(δ)dδ

= −τ 2 ln(δ)δτ−1 > 0. Therefore Z(δ) < 1 for δ ∈ (0, 1). Thus, ∂µ(s)∂λ

< 0, which in

turn implies 0 < c(s) < s as λ increases from λ = 0, as desired. �

A.9.2 Proof of Proposition 7, part (i)

For part (i), note E[τ ] is decreasing in λ (so that the contemporaneous expected con-

sumption frequency is increasing in λ) holding τ fixed. Also, E[τ ] is increasing in τ

holding λ fixed, where τ is decreasing in λ because τc > 0 and c(s) is decreasing in λ

from part (ii). Therefore, dE[τ ]dλ

< 0, implying that the expected frequency is increasing

in λ, as desired. �

A.10 Proof of Proposition 8

E[δτ(s,c)] = Pr(τ(s, c) < τ(s, 0))E[δτ(s,c)|τ(s, c) < τ(s, 0)]

+ (1− Pr(τ(s, c) < τ(s, 0)))E[δτ(s,c)|τ(s, c) ≥ τ(s, 0)]

As λ → ∞, Pr(τ(s, c) < τ(s, 0)) → 1, which implies E[δτ(s,c)] → E[δτ(s,c)|τ(s, c) <

τ(s, 0)]. This expectation is independent of c since its associated probability distri-

bution is the right-truncated exponential distribution on (0, τ(s, 0)) for all c. Note

since c = 0 minimizes τ(s, c), this means the probability of a craving-induced decision

converges to zero regardless of chosen consumption.

Now inspect the optimization problem given by the Bellman:

V (s) = −1 + maxc{u(c) + E[δτ(s,c)]V ((1−σ)s+ σc)}.

40

As λ → ∞, E[δτ(s,c)] > 0 converges to a constant and V ((1−σ)s + σc) falls with c.

Since u(c) is strictly decreasing for fixed θ > 0, it follows that

0 = limλ→∞

{arg max

c{u(c) + E[δτ(s,c)]V ((1−σ)s+ σc)}

},

which implies limλ→∞

c(s) = 0 for all s, as desired. �

A.11 Proof of Proposition 9

Let (c0, c1, . . .) denote the optimal consumption sequence given an availability schedule

T , U denote the lifetime utility associated with (c0, c1, . . .) given T is in fact the avail-

ability schedule, and (t0, t1, . . .) denote the decision point sequence that corresponds

to this situation. Let U∗ denote the lifetime utility associated with the optimal (i.e.

equilibrium) consumption sequence (c∗0, c∗1, . . .) in the original, cravings-only setting (in

which the good is always available and where availability is not tied to cues/decision

points), and (t∗0, t∗1, . . .) denote the corresponding decision point sequence. Now let U

denote the lifetime utility associated with the consumption sequence (c0, c1, . . .) from

before, except choosing it in the original, cravings-only setting, as opposed to the set-

ting with the availability schedule T ; that is, U is the lifetime utility if the agent

happens to choose (c0, c1, . . .) despite being in the original setting in which (c∗0, c∗1, . . .)

is optimal. Finally, let (t0, t1, . . .) denote the decision point sequence associated with

this “mismatched” scenario.

By optimality of (c∗0, c∗1, . . .) in the original setting, U∗≥ U . Now, for the consump-

tion sequence (c0, c1, . . .), the instantaneous return at decision point i is −1 + u(ci) —

regardless of whether the agent is in the original setting or the setting with availabil-

ity schedule T . Thus, the instantaneous return at decision point i is the same in the

situations characterized by U and by U . Now t0 = min{t0,min{T}} ≤ t0. That is,

the first decision point in the scenario characterized by U is the first decision point

in the original setting unless the good is available sometime before then (as deter-

mined by T ), in which case the first decision point is the earliest time that the good is

available (i.e. min{T}). Next, ti+1 − ti = min{τ(si, ci),min{T ∩ (ti,∞)} − ti}, while

ti+1 − ti = τ(si, ci), where si and si — defined as the decision-i habit stocks in the

U and U scenarios, respectively — are equal. That is, the interval in the scenario

characterized by U is the same as in the scenario characterized by U , unless the good

is available at a time between ti and ti + τ(si, ci) (non-inclusive), in which case the

interval is the difference between the earliest such time and ti. Thus, we have that

ti ≤ ti for all i. In turn, since the instantaneous decision-i returns are the same, it

41

follows that U ≥ U .

Now if the availability schedule is T ∗ = (t∗0, t∗1, . . .), then the lifetime utility asso-

ciated with the consumption sequence (c∗0, c∗1, . . .) in the new, access-as-cue setting is

simply U∗ because the decision point sequence is unaffected by the redundant cues

at (t∗0, t∗1, . . .). Therefore, the lifetime utility associated with T ∗ in the new setting

maximizes lifetime utility, i.e. U∗≥ U in general, where U∗= U if T = T ∗. �

A.12 Small σ as a Nonconstraint on τ and u

A.12.1 Formal Result and Interpretation

Proposition 10 Suppose Assumptions 1 and 2 hold for some initial parameterization,

(τ, u, σ′, δ′) with σ′, δ′ ∈ (0, 1). Then, for any σ′′ ∈ (0, σ′), there is a δ′′ ∈ (δ′, 1) such

that, for the new parameterization (τ, u, σ′′, δ′′):

(i) Assumptions 1 and 2 still hold,

(ii) s is a steady-state if and only if it was a steady-state under the initial parameter-

ization.

Since the new speed factor σ′′ can be any (suitably small) value less than the orig-

inal σ′ ∈ (0, 1), it follows from part (i) of Proposition 10 that the small σ restriction

does not constrain the domains for τ and u that the model can accommodate.65 An

implication of part (ii) is that users in behaviorally identical steady-states (same c∗

and τ ∗) with the same τ and u can be parametrically different, since c∗ and τ ∗ must

carry over to the new parameterization given that s∗ does. The implicit relationship,

δ(σ) =[1− σ

(τ∗sτ∗c

+ 1)]− 1

τ∗, which applies to both the initial and new parameteriza-

tions, allows us to compare two such individuals under ZP (see proof for derivation).

Observe that a more patient (higher δ) user has a habit that adjusts slower to con-

sumption (lower σ), which follows because both the discount and speed factors dampen

the long-term costs of consumption. Also note that any addicted steady-state will exist

for δ arbitrarily close to one, suggesting that the model accommodates realistic and

“rational” parameterizations of patience.66

65Note, ZP and NP would trivially carry over under the new parameterization since these conditions do notdepend on either the speed or discount factors.

66A caveat: near-perfect patience comes hand-in-hand with a colossally slow habit stock. A caveat to thecaveat: in the present construction, consumption has no persistent, positive effect beyond the next decisionpoint. This was a simplification to obviate the need for multiple state variables. By incorporating persistenceto the short-term effect of consumption, a given δ could coexist with a faster habit stock, all else equal.

42

A.12.2 Proof of Proposition 10

For ease of exposition, we will prove Proposition 10 for ZP. As before, although the

expressions become more complicated, it is straightforward to adapt what follows to

NP. First observe that parts (i)-(iii) of Assumption 1 do not depend on either σ or δ

and therefore still hold under the new parameterization. Given σ′′ < σ′, define

δ′′ =

[1− σ′′

(τ ∗sτ ∗c

+ 1

)]− 1τ∗

, (11)

where τ ∗, τ ∗s , τ∗c are evaluated at any (doesn’t matter which) steady-state under the

original parameterization given ZP. (The existence of such a s∗ is guaranteed because

Proposition 1 does not rely on the small σ restriction.) From µ(s∗) = 0, it follows that

δ′ =[1− σ′

(τ∗sτ∗c

+ 1)]− 1

τ∗ ∈ (0, 1). Thus σ′′ < σ′ implies δ′ < δ′′ < 1, which can be

seen from differentiating δ(σ) =[1− σ

(τ∗sτ∗c

+ 1)]− 1

τ∗with respect to σ.

To show that part (iv) of Proposition 1 holds under the new parameterization, first

note that concavity of −δτ requires α2 ∂2[−δτ(s,c)]∂s2

+ 2αβ ∂2[−δτ(s,c)]∂s∂c

+ β2 ∂2[−δτ(s,c)]∂c2

< 0, for

all s, c and for any α, β ≥ 0 except α = β = 0. This condition is equivalent to

α2[τss − ln(δ)τ 2s ] + 2αβ[τsc − ln(δ)τsτc] + β2[τcc − ln(δ)τ 2

c ] < 0.

Note that τsc − ln(δ)τsτc is proportional to the cross partial derivative of the effective

payoff function. Since δ′ < δ′′ < 1 implies − ln(δ′)τsτc < − ln(δ′′)τsτc, we have τsc −ln(δ′′)τsτc > τsc − ln(δ′)τsτc > 0. Therefore −∂2δτ(s,c)

∂s∂c> 0 for all s, c under the new

parameterization.

Now suppose concavity is violated under the new parameterization. Since τss −ln(δ)τ 2

s and τcc − ln(δ)τ 2c are both decreasing in δ and are both negative for δ = δ′

(follows from setting α = 1, β = 0 and α = 0, β = 1, respectively), both expressions

are still negative for δ = δ′′ > δ′. Thus, for concavity to be violated, we must have

(fixing α = 1, without loss of generality):

[τss − ln(δ′′)τ 2s ] + 2β[τsc − ln(δ′′)τsτc] + β2[τcc − ln(δ′′)τ 2

c ] > 0,

for some β > 0. (Here, β > 0 follows because concavity is assured with β < 0

since the cross partial derivative is positive.) Given − ln(δ′′) < − ln(δ′), we now need

0 > τ 2s + 2βτsτc + β2τc = (τs + βτc)

2 > 0, which is a contradiction. Thus concavity

holds under the new parameterization.

For Assumption 2, from the definition of µ, rearranging Equation (11) gives µ(s|δ′, σ′) =

43

0 if and only if µ(s|δ′′, σ′′) = 0, implying that the steady-states carry over. Now rewrite:

µ(s, σ) =

[τc + σ(τs + τc)

[δ(σ)τ ]

1− [δ(σ)]τ

]c=s

,

where δ(σ) =[1− σ

(τ∗sτ∗c

+ 1)]− 1

τ∗. Now µ(s, σ′) 6= 0 implies µ(s, σ) 6= 0 for all σ ∈

[σ′′, σ′] due to the steady-state equivalence under δ(σ). It follows from continuity of

µ(s, σ) in σ that µ(s, σ′) ≶ 0 if and only if µ(s, σ′′) ≶ 0, which in turn implies that

parts (i)-(iii) of Assumption 2 are preserved under the new parameterization. �

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