Bad Habits and the Endogenous Timing of Urges*

PETER LANDRY

University of Toronto

November 12, 2017

I present a theory of harmful addiction in which “giving in” to an unwanted urge (i.e.

consumption) delays the recurrence of urges in the short-run, but increases their long-run

frequency. The theory offers new predictions as to how the frequency, levels, cue-dependence,

and temporal consistency of consumption evolve during habituation, while uniquely capturing

near-term substitution in demand across time. New welfare implications for restrictions on

consumption and on marketing are also addressed.

** Philipp Sadowski provided excellent guidance throughout this project. Philipp Kircher and threeanonymous referees provided excellent feedback which enormously improved the paper. I am also greatlyappreciative of very helpful feedback from Peter Arcidiacono, Attila Ambrus, Jonas Arias, Colin Camerer,Michael Dalton, Rahul Deb, Tanjim Hossain, Joseph Hotz, Rachel Kranton, Mehmet Ozsoy, Curt Taylor,and Huseyin Yildirim.

1

1 Introduction

It is well-known that many addictive substances are harmful to those who consume them.

Smoking alone is responsible for an estimated 6 million deaths each year. Nonetheless,

nearly one billion people worldwide continue to smoke on a daily basis (Ng et al., 2014).

Such habitual consumption behaviors have long been linked to the high frequency of

urges that arise in an addiction (e.g. West and Schneider, 1987; Flannery et al., 1999).

But what exactly is an urge? And how is it that an urge could motivate the consumption

of cigarettes, let alone harder drugs, such as heroin or cocaine?

This paper presents a theory of addiction based on the so-called stubbornness and

reinforcement of unwanted urges. Stubbornness refers to the idea that choosing to resist

an urge (i.e., abstaining) can invite continued urges for some period of time. This aspect

can motivate consumption as a means to satisfy the urge, delaying the recurrence of

unwanted urges in the short-run. Reinforcement describes an opposing long-run effect

whereby urge-induced consumption will, by virtue of strengthening the habit, eventually

lead to a higher frequency of urges. Through repeated reinforcement over time, the

relatively mild urges experienced before initial consumption (manifested as an occasional

curiosity or peer pressure, perhaps) can give way to the more prominent urges, namely

frequent biological cravings, characteristic of full-fledged addictions.1

The theory operationalizes stubbornness and reinforcement by allowing the times at

which urges arise to depend on past consumption choices. A basic, deterministic version

of the “endogenous timing” (ET) model is first analyzed in Section 2, and is then extended

in Section 3 to include stochastic environmental cues, i.e. external stimuli — such as

someone else smoking or a cigarette advertisement — that can trigger an urge to consume.

The behaviors predicted by the ET model capture a number of empirical patterns that

are not addressed by prevailing theories of addiction. Below we discuss these distinctions

(summarized in Table 1) in relation to four leading theories: Becker and Murphy’s (1988)

standard “rational addiction” theory based on habit-formation preferences, in which the

marginal utility of consumption — and thus, the incentive to consume — rises with

past consumption;2 Gul and Pesendorfer’s (2007) theory, in which addiction is driven

by temptation costs (from not consuming the addictive good) that grow with past con-1 Illustrating the stubbornness of cravings, Koob and Le Moal (2008) note addicts’ “intense cravings

during abstinence,” while Bell et al. (1999) observe that “smoking decreases tobacco craving.” Reportedas the most frequently-cited motive for trying cigarettes (Milton et al., 2008) and e-cigarettes (Pepper etal., 2014), curiosity can be considered stubborn because it too tends to persist until satisfied (Loewen-stein, 1994). Insistent peer pressure is another commonly-cited reason for starting smoking (Evans etal., 1978; de Vries et al., 2003), and may also function as a stubborn urge in this way.

2 Discussions of rational addiction theory’s behavioral predictions will implicitly refer to both theBecker-Murphy model and Gruber and Koszegi’s (2001) adaptation with present-biased discountingbecause the models generate qualitatively identical behaviors in standard dynamic choice settings (inwhich precommitment is unavailable).

2

sumption; Laibson’s (2001) theory, which features habit-formation preferences analogous

to those in rational addiction theory, except they are activated at random times by ex-

ogenous environmental cues; and Bernheim and Rangel’s (2004) theory of addiction as a

growing vulnerability to stochastic environmental cues that trigger “unintentional” (i.e.

suboptimal) consumption of the addictive good.

(I) Observable Measures of Habit Strength: Consumption Levels and Frequency. Em-

pirical research shows that stronger consumption habits are behaviorally characterized by

both higher consumption frequencies and higher levels of consumption per occasion. For

instance, smokers who smoke more frequently also spend more time puffing each cigarette

(Fagerstrom and Bates, 1981), inhale more nicotine per cigarette (Shiffman, 1989), and

leave shorter unsmoked cigarette butts after smoking (Shiffman et al., 1994). Consistent

with the evidence, the ET model predicts that the frequency and the per-occasion lev-

els of consumption both rise as a weaker habit develops into an addiction.3 Prevailing

addiction theories capture (at most) one of these two aspects of habit strength, as the

rational addiction and Gul-Pesendorfer temptation theories predict rising per-occasion

consumption levels during habituation but no change in frequencies, while consumption

levels do not change in the Laibson and Bernheim-Rangel cue theories.

(II) Adjacent Substitution and Distant Complementarity. Empirical research shows

that depriving smokers of cigarettes leads to subsequent increases in the total number

of cigarettes smoked over relatively short time frames — on the order of five hours

(Zacny and Stitzer, 1985) to one week (Erskine et al., 2010). Such findings illustrate

adjacent substitution in the demand for addictive goods, i.e. present demand and demand

in the near-term future tend to be substitutes.4 Another empirical regularity, distant

complementarity, means present demand and demand in the long-run future tend to

be complements, as revealed by the positive relationship in the demand for cigarettes

between two consecutive years (Chaloupka, 1991; Becker et al., 1994). Accounting for

both regularities, the ET model predicts that present and future demand for the good

will exhibit substitution in the short-run but complementarity in the long-run. While

3 This prediction could be re-stated simply as saying those who consume more frequently also choosehigher consumption levels (i.e. without specifying “as a weaker habit...”), with either the frequency orlevels of consumption implicitly understood (here and throughout) as our empirical definition of habitstrength. That said, neither measure provides a universal definition that can be used to classify thepredictions of other theories considered here. For example, we cannot meaningfully ask (as in item III)how the ‘cue-dependence’ varies with the levels of consumption in a theory for which consumption levelsdo not vary. Thus, when classifying each theory’s predictions, we instead define habit strength by theassociated state variable (present in each theory) that reflects a time-weighted stock of past consumption.Habit duration may also be understood as a proxy for habit strength (in light of evidence that dailycigarette use tends to rise gradually over years of smoking; see Chassin et al., 1996).

4 Also see Evans et al.’s (1999) finding that the number of cigarettes smoked per day falls by lessthan the share of time during the workday when a workplace smoking ban is instated, implying smokersconsume more cigarettes outside working hours in response to the ban.

3

other theories predict distant complementarity too, they do not (simultaneously) predict

its opposing short-run effect. In fact, adjacent complementarity — which is the opposite

of adjacent substitution — is commonly regarded as the defining property of addiction

(beginning with Becker and Murphy, 1988).

(III) Stronger Cue-Dependence for Weaker Habits. Empirical studies consistently show

that consumption is more strongly associated with environmental cues for those with

weaker consumption habits. As one example, Shiffman et al. (2014) find the extent to

which the presence of someone else smoking increases one’s propensity to smoke is 2 to

3 times larger for occasional smokers than for daily smokers. Cronk and Piasecki (2010)

similarly study the effect of someone else smoking (among several other environmental

factors), reporting “less cue control over smoking for daily than nondaily smokers.”5

Capturing this relationship, the ET model predicts that the dependence of consumption

on environmental cues (as measured by the proportion of consumption that coincides with

a cue) decreases as a weaker habit develops into an addiction. In contrast, the Bernheim-

Rangel theory predicts the opposite relationship, as consumption occurs independently of

environmental cues during (but not after) the “casual user” phase that precedes addiction,

while in other theories consumption is either entirely dependent or entirely independent

of environmental cues regardless of habit strength.

(IV) Formation of Consistent Consumption Routines in Addiction. A related line of

research reveals that, in contrast to occasional users’ relatively sporadic consumption

patterns, addicts tend to develop regimented consumption routines featuring consistent

time-intervals between consumption occasions (Benowitz, 1991; Shiffman et al., 2004).

Accordingly, the ET model (with cues) predicts that consumption schedules become more

consistent (as reflected by a lower variance in the time between consumption occasions) on

the path to addiction. The Bernheim-Rangel theory instead predicts the reverse pattern,

while other theories do not allow the variability (or lack thereof) of consumption schedules

to vary with habit strength.

In addition to its descriptive predictions, the ET model also offers new implications for

public policy (see Table 2) and for individual treatment. For instance, policies that reduce

environmental cues (such as an advertising ban) are best-suited for helping nonusers and

users with weaker habits, while those with stronger habits disproportionately benefit from

interventions to reduce internal cravings (such as wearing a nicotine patch). As discussed5 In the psychology literature, the relative independence of consumption on environmental cues is often

considered a defining feature of addiction (Shiffman et al., 2004; Shiffman and Paty, 2006). Such notionsare also in line with evidence of a disproportionate effect of advertisements on younger smokers (who tendto have less-developed habits and consume less frequently). For example, Pollay et al. (1996) estimateteenagers’ sensitivity to cigarette advertisements is roughly triple that of adults, while Cummings et al.(1997) report that over 90 percent of adolescent smokers (versus 35-40 percent of adults) smoke one ofthe top-three selling brands — which are also the most heavily advertised — while generic brands usingminimal advertising capture a five-fold greater share of adult than adolescent demand.

4

in Section 4, this characterization fits with the implicit tailoring of nicotine replacement

therapies to heavy smokers as well as the targeting of policies that reduce cues towards

potential new users. The Laibson and Bernheim-Rangel cue theories also accommodate

the possibility of welfare gains from policies that reduce environmental cues, but imply

that nonusers lacking any habit would not benefit from such policies.

In the absence of externalities, the ET model does not provide a normative justification

for policies that restrict consumption. However, the model naturally implicates a negative

externality from public consumption in that it exposes others to unwanted cues. In this

vein, the analysis of public consumption bans reveals a strict benefit to nonusers and users

with sufficiently weak habits, while those with stronger habits are hurt. Consistent with

this, Green and Gerken (1989) find that opposition to such restrictions increases with

habit strength, as 16% of nonsmokers, 32% of light smokers, and 62% of heavy smokers

would support weakening existing restrictions on smoking; other studies similarly show

that heavy smokers — but not light smokers — tend to oppose bans on smoking at

restaurants (Brooks and Mucci, 2001) and their place of work (Daughton et al., 1992).6

In contrast, the Laibson and Bernheim-Rangel cue theories — which also formalize the

notion that consumption restrictions could entail fewer environmental cues — imply that

policies of this sort would provide the most benefit to addicts while failing to help those

lacking any such habit.

Table 1. Behavioral ImplicationsET RA GP-T L-C BR-C

(I)how consumption ‘amounts’ changeas weaker habit turns into addiction

frequency

levels

↑

↑

0

↑

0

↑

0

0

↓

0

(II)present demand & future demand:(C)omplements or (S)ubstitutes?

short-run

long-run

S

C

C

C

C

C

C

C

C/S

C/S

(III)how dependence of consumption on environmentalcues changes as weaker habit turns into addiction

↓ 0 0 0 ↑

(IV)how variability of consumption schedules

changes as weaker habit turns into addiction↓ 0 0 0 ↑

Behavioral predictions of the ET model, compared to rational addiction theory (RA), Gul-Pesendorfer tempta-tion theory (GP-T), Laibson cue theory (L-C), and Bernheim-Rangel cue theory (BR-C). For clarity, we restrictour attention to specifications of the Bernheim-Rangel theory that adhere to their premise that addiction entailscue-induced consumption “mistakes.” For an explanation of each item, see Appendix E. For clarification onhow habit strength is defined in I, III, and IV, see footnote 3.

6 These patterns could also be explained (at least in part) by a desire among nonusers and lightusers to avoid the smell and/or health costs of second-hand smoke. While important, direct negativeexternalities such as these are not addressed by the ET model.

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Table 2. Welfare Impacts

habit status ET BM-RA GK-RA GP-T L-C BR-C

advertising bannone

addict

↑

↑

0

0

0

0

0

0

0

↑

0

↑

private consumption bannone

addict

0

↓

0

↓

0

↑/↓

↑

↑

0

↓

0

↑

public consumption bannone

addict

↑

↓

0

↓

0

↑/↓

↑

↑

0

↑

0

↑

Welfare predictions, compared to prevailing addiction theories. Here we distinguish between the Becker-Murphy (BM-RA) and Gruber-Koszegi (GK-RA) versions of rational addiction theory.

2 Basic Model

An individual experiences recurring urges at times t0, t1, t2 . . .. At each urge, the individ-

ual chooses a consumption level c ∈ [0,1] and receives a direct (dis)utility u(c), where

ci will denote the consumption level at ti. We assume consumption is harmful in that

u′(c) < 0 and interpret this as encompassing both monetary and non-monetary (e.g.

health-related) consumption costs net of any pleasurable effects. We also assume urges

are unwanted with a fixed ‘urge cost’ u(0) = −1 (normalized for simplicity) and interpret

this as encompassing any inherent hedonic and/or attentional costs when confronting an

urge.7,8

Next, the habit stock s ∈ [0,1] is defined as a time-weighted average of past consump-

tion levels that evolves according to

si+1 = (1 − σ)si + σci, (1)

where si and si+1 denote the habit stocks at ti and ti+1 (respectively) and the “habitua-

tion rate” σ ∈ (0,1) determines how quickly the habit stock changes with consumption.

Observe that choosing ci = si implies si+1 = si, which means the consumption level and

the habit stock will be equal in a steady-state.

7 An attentional cost may reflect an urge’s capacity to disrupt attention to an ongoing activity, forcingthe individual to “think about” consumption. In addition to their hedonic symptoms (e.g. headaches),biological cravings are known to disrupt concentration on other tasks (Hughes and Hatsukami, 1986;DiFranza and Wellman, 2005). Psychology research (reviewed by Loewenstein, 1994) has also implicatedboth forms of costs as features of curiosity (conceived here as a potential urge arising prior to habitua-tion), describing its “painful feelings” if unsatisfied as well as its demands on attentional resources.

8 See Appendix D for a behaviorally-equivalent (though less parsimonious) representation of the utilityfunction, which can be interpreted as allowing: (i) the magnitude of the urge cost to grow with pastconsumption; (ii) the marginal utility of consumption to increase with past consumption; and (iii) themarginal utility of consumption to be positive (for small c).

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The length of time between consecutive urges is then determined by

ti+1 − ti = τ(si, ci), (2)

where τ is called the interval function. Using subscripts to denote partial derivatives, we

assume τc(s, c) > 0 to represent the stubbornness of urges, so that increasing consumption

delays the next urge. In turn, we assume τs(s, c) < 0 to represent the reinforcement of

urges, so that stronger habits entail more frequent urges, all else equal. Since urges are

costly, the stubbornness property τc(s, c) > 0 creates an incentive to consume as a means

to delay the next urge. However, since consumption causes the habit stock to grow

through (1), the reinforcement property τs(s, c) < 0 provides a disincentive to consume

(in addition to the direct disincentive from u′(c) < 0) as consumption leads to a higher

frequency of urges in the long-run.

Instead of working with τ directly, we will often work with the effective discount

function D(s, c) ≡ e−rτ(s,c), so that D(si, ci) represents the time-ti discount on utility at

ti+1 given the subjective discount rate r > 0. For cleanliness and without loss of generality,

we normalize r = 1, which means time is implicitly expressed in the unit for which the

discount over one unit is 1/e (for example, if rd is the daily discount rate, fixing r = 1

effectively defines one time-unit as r−1d days).

Note, given the present consumption level c and habit stock s, the discounted urge cost

from the next urge is simplyD(s, c)u(0) = −D(s, c). Since delaying the next urge provides

the incentive to consume, −Dc(s, c) can therefore be understood as the (marginal) benefit

or “motivation” to increase consumption from c at s.

2.1 Dynamic Optimization

The optimization problem is expressed recursively through the Bellman equation:

V (s) = maxc

{u(c) +D(s, c)V ((1 − σ)s + σc)}, (3)

where V (s) represents the value function at the time of an urge given the current habit

stock s.9 Here it is implicitly presumed that the individual never consumes in the absence

of an urge. This feature can be motivated by a notion that the stubbornness of urges is

not in effect when an urge is not present (while consumption remains harmful), implying

9 We may notice that the dynamic optimization problem in (3) is equivalent (thus generating the sameoptimal consumption levels) to that of a discrete-time (t = 0,1 . . .) model in which D(s, c) representsan endogenous discount factor between consecutive periods (as opposed to its present use as embeddingan endogenous time-interval to the next urge/decision). With that said, a discrete-time, endogenousdiscounting model of this sort would still not be behaviorally equivalent to the present model becausethe models would generate different predictions as to when consumption occurs.

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abstinence would not expedite the next urge — thus eliminating any incentive to consume

— at such times. As an alternate motivation, the individual may not even consider the

consumption decision unless triggered by an urge to do so.

Using (3), we can compute the first-order and envelope conditions (respectively):

0 = u′(c) +Dc(s, c)V ((1 − σ)s + σc) + σD(s, c)V ′((1 − σ)s + σc), (4)

V ′(s) = Ds(s, c)V ((1 − σ)s + σc) + (1 − σ)D(s, c)V ′((1 − σ)s + σc). (5)

An interior optimal consumption path, if one exists, will then satisfy the Euler equation:

(1 − σ)D(si, ci)u′(ci+1) − u

′(ci) −Dc(si, ci)u(ci+1)

Dc(si, ci)D(si+1, ci+1) + (σDs(si+1, ci+1) − (1 − σ)Dc(si+1, ci+1))D(si, ci)− u(ci+2)

=((1 − σ)D(si+1, ci+1)u

′(ci+2) − u

′(ci+1) −Dc(si+1, ci+1)u(ci+2))D(si+2, ci+2)

Dc(si+1, ci+1)D(si+2, ci+2) + (σDs(si+2, ci+2) − (1 − σ)Dc(si+2, ci+2))D(si+1, ci+1).

(6)

A derivation of (6) from (3)-(5) is provided in Appendix A.

2.2 Analysis

To facilitate further analysis of the model, we adopt functional forms for u and τ .10 In

particular, we first assume that the utility function consists of a linear consumption cost

in addition to the fixed urge cost:

u(c) = −(1 + θc). (7)

Here the “direct cost” parameter θ > 0 can be understood as a reduced-form represen-

tation of the multiple channels — including monetary as well as non-monetary costs —

through which consumption can be costly in and of itself (i.e., not including the indirect

future consequences arising through habituation).

Next, we assume the interval function is comprised of additively-separable logarithmic

components as follows:11

τ(s, c) = ln(γ + 1 + c) − ln(γ + s). (8)

We can see this satisfies the stubbornness and reinforcement properties, with τc(s, c) =

(γ + 1+ c)−1 > 0 and τs(s, c) = −(γ + s)−1 < 0, while the magnitudes of τc(s, c) and τs(s, c)

10 The Euler equation (6) includes functions (in this case, u and τ) corresponding to three consecutivedecisions, which is one more than typically seen in such conditions. This added complexity stems fromthe nonstandard feature of the Bellman in (3) whereby the choice of c affects the discounted future valueterm through D(s, c) in addition to its effect on V ((1 − σ)s + σc).

11 The mathematical convenience of a logarithmic form results from the fact that τ only enters theEuler equation (6) through D = e−τ .

8

are both decreasing in γ > 1 (suggesting γ “dampens” these properties). The inclusion

of +1 in the first term, ln(γ + 1 + c) > 0, but not the second term, − ln(γ + s) < 0, ensures

τ(s, c) ≥ 0 for all s and c as well as ∣τs(s, c)∣ ≥ τc(s, c) (with both inequalities binding only

in the special case with c = 0 and s = 1). This latter inequality suggests the reinforcement

of urges is stronger (in a sense) than the stubbornness. For example, ∣τs(s, s)∣ > τc(s, s)

implies ddsτ(s, s) = τs(s, s) + τc(s, s) < 0, so that choosing c = s (which will hold in a

steady-state) entails a shorter time between urges when s is larger — despite requiring

a larger c.

Unlike the interval function, the effective discount function implied by (8), D(s, c) =γ+sγ+1+c , is not additively separable. Of particular importance, its cross-partial derivative

is negative, Dsc(s, c) = −(γ + 1 + c)−1 < 0, which implies the motivation to consume (as

captured by −Dc(s, c) > 0) is greater when s is large.12 In this way, urges become more

“powerful” through reinforcement — in addition to being more frequent — even though

the −1 urge cost and the extent to which consumption delays the next urge (τc(s, c) > 0)

are both independent of s.13

2.3 Steady-States

Letting c∗(s) denote the optimal consumption level given s, a steady-state is defined as

a value of s for which c∗(s) = s. By inserting our functional forms for u and τ into the

Euler equation (6) and setting all consumption and habit stock variables to s, we can

derive and succinctly express the condition for s ∈ (0,1) to be an interior steady-state as:

θ =(1 − σ)(γ + s)

1 + (γ + s)(1 + σ(γ + 1) − (1 − 2σ)s). (9)

It is readily verifiable that up to two interior steady-states may coexist, as (9) can be

rearranged as a quadratic in s.

Our first result identifies conditions under which three distinct steady-states will exist

(one of which is a corner solution), with each offering a stylized representation of a

relevant population subgroup in the context of smoking.

12 Intuitively, this follows from the fact that (due to discounting) the benefit in present value fromdelaying a distant future cost is less than from delaying a temporally close future cost (of equal mag-nitude) by the same amount of time — and with τs(s, c) < 0, the next urge is in fact closer when s islarger.

13 As alluded to in footnote 8, Appendix D describes a reformulation of the utility function underwhich urges can be interpreted as “more powerful” for larger s in two additional ways. Namely, themagnitude of the urge cost and the marginal utility of consumption both increase with s. Since thislatter property is the defining feature of the habit-formation preferences in rational addiction theory(Becker and Murphy, 1988), the basic ET model (without cues) could, in principle, be understood asintegrating the endogenous timing mechanism through τ with rational addiction-style preferences. Asdiscussed in the appendix, however, these preferences are extraneous because the reformulated model isbehaviorally-equivalent to the model with u as conceived here.

9

Proposition 1 Consider the following candidate steady-states:

● Nonuser: a stable steady-state at s = 0;

● Chipper: an unstable steady-state at some sL ∈ (0, 12);

● Addict: a stable steady-state at some sH ∈ (12 ,1).

There exist σ, σ, θ, θ with 0 < σ < σ < 1 and 0 < θ < θ such that all three candidate

steady-states will exist, with si converging to sH if s0 ∈ (sL,1] and to zero if s0 ∈ [0, sL),

provided σ ∈ [σ,σ] and θ ∈ (θ, θ).

All proofs are in Appendix B. While omitted from the main text for cleanliness, closed-

form expressions for the steady-state habit/consumption levels sL and sH , the corre-

sponding time-intervals between consumption occasions, τ(sL, sL) and τ(sH , sH), and

the parametric bounds σ, σ, θ, and θ (which depend on γ) are provided in Appendix C.

Proposition 1 establishes three steady-states of interest, ranging from the nonuser

steady-state at s = 0, which can be thought to represent a nonsmoker, to the addict

steady-state at sH . There is also an intermediate steady-state at sL representing a mem-

ber of the subgroup of smokers — often called chippers — who only smoke occasionally.

The instability of sL means “approximate” chippers in the vicinity of sL will tend to drift

away from sL, either towards 0 or sH , which qualitatively fits with evidence of their tran-

sience as chippers (in comparison to nonsmokers and regular smokers) generally maintain

their status as such for relatively short periods of time (Zhu et al., 2003). In this way, the

chipper steady-state can also be thought of as a “tipping point” for addiction, as those

with s > sL (but not those with s ≤ sL) converge to sH in the long-run.

Since s = 0 is a steady-state, first-time consumption and the possibility of becoming

addicted may be generated by an exogenous shock to the habit stock (as in Becker and

Murphy, 1988) or by allowing the initial habit stock to be positive (as in Laibson, 2001).

As an alternate mechanism that does not require us to allow s > 0 before consumption

has occurred, the online appendix describes how initial consumption (possibly leading to

addiction) could be motivated by a temporary decrease in θ > 0 or by a misconception

regarding its magnitude.

Lastly, the restrictions θ ∈ (θ, θ) and σ ∈ [σ,σ] in Proposition 1 can be understood from

the fact that both parameters signify costs of consumption — whether direct through θ

or indirect through σ (from faster habituation). If either type of cost is too large, then

c∗(s) < s for all s ∈ (0,1], precluding a steady-state with positive consumption. If either

is too small, however, then c∗(0) > 0, precluding a zero-consumption steady-state (which

presumably describes most people). From here on, we assume θ ∈ (θ, θ) and σ ∈ [σ,σ],

except where otherwise noted.

10

2.4 Measures of Habit Strength: Levels and Frequency

By comparing the steady-state consumption patterns of an addict and a chipper, we can

verify that the stronger habit entails higher per-occasion consumption levels, sH > sL, as

well as shorter intervals between consumption occasions, τ(sH , sH) < τ(sL, sL) (equiv-

alently, a higher consumption frequency). Recalling our discussions in Section 2.2, the

addict’s higher consumption levels can be understood as reflecting a higher motivation

to consume (as implied by Dsc(s, c) < 0), while the addict’s higher consumption fre-

quency reflects the property that reinforcement outweighs stubbornness in the sense that

∣τs(s, c)∣ > τc(s, c).

The next result extends this dual concept of habit strength beyond our comparison of

these steady-states.

Proposition 2 As s rises, c∗(s) increases and τ(s, c∗(s)) decreases.

As discussed in the introduction, the prediction that the frequency and per-occasion levels

of consumption both increase as a habit strengthens fits with evidence that more frequent

smokers spend more time puffing each cigarette, inhale more nicotine per cigarette, and

leave shorter unsmoked cigarette butts after smoking. Prevailing addiction models, how-

ever, do not simultaneously account for both aspects of habit strength (see Table 1).

2.5 Intertemporal Demand Relationships

We now examine the relationship between present and future demand for the good. To

do this, we let N `(c ∣s) denote the number of urges occurring within a length-` time-

window on the optimal consumption path after consuming c (given s). Presuming future

consumption levels are positive, N `(c ∣s) also represents the number of consumption oc-

casions during this time.

Proposition 3 For all s, there exists a `0 such that N `(c ∣s) is decreasing in c for all

` < `0, and increasing in c for all ` > `0.

The short-run effect in Proposition 3 (for ` < `0) captures evidence of adjacent substitu-

tion in the demand for cigarettes, as demonstrated by findings that temporarily depriving

a smoker of cigarettes leads to subsequent increases in the total number of cigarettes con-

sumed over relatively short time frames.14 In turn, the long-run effect captures evidence

14 The implied mapping between the number of future cigarettes consumed and N `(c ∣s) is reasonable if

smokers vary consumption levels by varying the amount of nicotine inhaled per cigarette. In other cases,it may be more reasonable to quantify future demand in terms of (time-aggregated) consumption levels.For instance, demand among e-cigarette smokers (who typically re-use the same e-cigarette) would morenaturally be measured by the volume of “e-liquid” consumed. The online appendix shows how adjacentsubstitution and distant complementarity can also be captured with demand defined in this way.

11

of distant complementarity, as demonstrated by a positive relationship in the demand

for cigarettes between two consecutive years. As discussed in the introduction, other

addiction theories predict distant complementarity as well, but do not simultaneously

capture adjacent substitution.

3 Stochastic Environmental Cues

Environmental cues are well-known in psychology research to elicit urges, leading to the

consumption of addictive goods.15 Environmental cues are also central to the economic

theories of addiction by Laibson (2001) and Bernheim and Rangel (2004), both of which

emphasize their stochastic nature in that cue-induced consumption tends to be unpre-

dictable, and driven by seemingly random external influences.

We now incorporate stochastic environmental cues into the ET framework, where

cues are assumed to arrive at a fixed rate λ > 0 and to elicit an urge upon arrival. The

“natural” interval function τ defined in (8) from the basic model now represents the

time between consecutive urges if no cues arise in the interim. The true interval ti+1 − ti

is therefore an exponential random variable parameterized by λ and right-censored at

τ(si, ci).

Lemma 1 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(s, c) ≡ ln(1+λ)−

ln(λ+ (γ+sγ+1+c)

1+λ). Furthermore, nonuser, chipper, and addict steady-states will continue

to exist (with their features as described Proposition 1), provided λ > 0 is not too large.

Hence, the optimization problem with stochastic cues is the same as in the basic, de-

terministic model using τ 0 as defined in Lemma 1. This equivalence allows us to carry

over previously-established equilibrium properties to the new setting, as long as the cue-

arrival rate λ > 0 is not too large. For this reason, we take λ > 0 to be sufficiently small

in our analysis.

Proposition 4 The probability that consumption coincides with a cue decreases with s.

This result captures previously-cited evidence that consumption is less dependent on

environmental cues for those with stronger habits.16 The effect can be understood as

15 See Caggiula et al. (2001) and Carpenter et al. (2009) for reviews. While this psychology literaturehas treated the sight of someone else consuming as the classic illustration of a cue, accumulating evidencesuggests advertisements (a standard example of a cue in the economics literature) can also elicit “urges,”whether by stimulating curiosity among potential new users (Pierce et al., 2005; Portnoy et al., 2014)or by inducing cravings among addicts (Paynter and Edwards, 2009; Vollstadt-Klein et al., 2011).

16 Though some cues may be hard to observe, the result would still hold if limited to a particulartype of readily identifiable cue. For instance, Proposition 4 would imply (all else equal) that the share

12

a consequence of reinforcement, τs(s, c) < 0, which diminishes the role of environmental

cues (relative to internal urges) in driving consumption behavior. As discussed in the

introduction, prevailing theories do not capture this relationship, with most predicting

that the association between consumption and environmental cues does not vary with

habit strength (see Table 1).

Proposition 5 The variance of the time-interval between consumption occasions de-

creases with s.

Thus, consumption patterns become more predictable as a weaker habit develops into

an addiction. This too can be understood as a consequence of reinforcement, as it en-

hances the role of deterministic (i.e. internal urges) relative to stochastic (environmental

cues) drivers of consumption. The result also matches evidence that more habituated

smokers (who have been smoking for a longer time and consume more cigarettes per day)

tend to develop relatively predictable consumption routines with consistent time-intervals

between consumption occasions (Benowitz, 1991; Shiffman et al., 2004). In contrast, pre-

vailing addiction theories do not allow the variability of consumption schedules to vary

with habit strength, with the exception of the Bernheim-Rangel theory, which instead

predicts that consumption patterns become less consistent as a weaker habit develops

into an addiction.

4 Implications for Policy and Individual Treatment

We now consider the (demand-side) consumer welfare implications of various approaches

to address addiction, including both policy-based and individually-pursued approaches.

The welfare impact of a particular approach will be measured by its effect on the same

objective function, i.e. V , that the individual seeks to maximize through consumption

choices. For a policy that only affects V through the choice variable c (as in our first result

below), the welfare impact will therefore correspond to choice behavior in a standard and

straightforward manner. In other cases, welfare-improving policies can be interpreted

(following Gul and Pesendorfer, 2007) as policies the individual would choose — i.e.,

vote in favor of — if given the option.

4.1 Bans on (Private) Consumption

We first consider a ban on consumption that permanently imposes c = 0. We refer to this

as a “private” consumption ban because, unlike our consideration of public consumption

of consumption that coincides with someone else smoking rises with habit strength (which itself may bemeasured by consumption frequency, consumption levels, or even duration of the habit — see footnote3). See Shiffman et al. (2014) and Cronk and Piasecki (2010) for evidence to this effect.

13

restrictions later in this section, the present exercise does not allow any externalities from

consumption.

Proposition 6 A private consumption ban is strictly welfare-reducing for all s with

c∗(s) > 0 and welfare-neutral for all s with c∗(s) = 0.

As in Becker and Murphy (1988) and Laibson (2001), a private consumption ban merely

constrains the optimal consumption choice, thus reducing welfare except for those who

would abstain regardless. In contrast, a private consumption ban may improve consumer

welfare by counteracting present bias in Gruber and Koszegi (2001), by reducing temp-

tation costs in Gul and Pesendorfer (2007), or by preventing consumption “mistakes” in

Bernheim and Rangel (2004).

4.2 Mitigation Approaches

Next we consider three forms of mitigation that, with one exception, are better under-

stood as individually-pursued measures than as feasible policy instruments. The first

is harm-reduction, formalized as a decrease in the direct cost parameter θ. Perhaps

the most well-known harm-reduction measure is switching from traditional cigarettes to

e-cigarettes, which contain nicotine but lack tar and other toxins found in traditional

cigarettes. Next is cue-reduction, formalized as a decrease in the cue-arrival rate λ.

Cue-reduction may be brought about by policies that restrict advertisements and other

promotions, or by individual actions taken to avoid cue-laden environments. Lastly,

we consider cravings-reduction, which is modeled by letting the time between consecu-

tive urges be a weighted average of the interval functions with and without cravings,

πτ(s, c) + (1 − π)τ(0, c), where π < 1 represents the extent to which cravings persist un-

der cravings-reduction. In practice, cravings-reduction can be achieved through nicotine

replacement therapies, such as wearing a nicotine patch, which are known to reduce the

frequency of urges (Foulds et al., 1992).

Lemma 2 Harm-reduction is strictly welfare-improving for all s with c∗(s) > 0 and

otherwise welfare-neutral; cravings-reduction is strictly welfare-improving for all s > 0

and welfare-neutral for s = 0; cue-reduction is strictly welfare-improving for all s ∈ [0,1].

Lemma 2 affirms that all three mitigation approaches do in fact improve welfare. How-

ever, this result should not be interpreted too strongly when considering individually-

pursued mitigation measures because the analysis abstracts from the costs inherent in

their use (as examples, switching from cigarettes to e-cigarettes requires the purchase of

a vaporizer or starter kit, while nicotine replacement therapies are generally more ex-

pensive than cigarettes). With that said, Lemma 2 can more plausibly be taken at face

14

value when considering its implications for cue-reduction policies, such as advertising

bans, which lack an obvious direct cost to consumers.

The general feature that cue-reduction policies can improve welfare is shared by the

Laibson and Bernheim-Rangel cue theories. However, Lemma 2’s implication that cue-

reduction strictly improves welfare even at s = 0 differs from these theories, which suggest

cue-reduction would have no welfare impact at their analogous zero-habit states.

Given that individuals with positive consumption can benefit from all three forms

of mitigation, we now turn to the question of targeting: who is best served by each

mitigation approach, or as formalized here, how do the relative welfare impacts of each

approach differ between a chipper and an addict?

In general, a chipper and an addict may both prefer (i.e. benefit more from) one

mitigation approach over another because the size of their welfare benefits do not solely

depend on the strength of the individual’s habit — they also depend on the extent to

which the associated parameters (θ, λ, or π) decrease. For example, if harm-reduction

provides a relatively large decrease in the direct cost parameter θ while cravings-reduction

provides a relatively small decrease in the ‘craving persistence’ parameter π, an addict

and a chipper would both trivially benefit more from harm-reduction than from cravings-

reduction. To assess the relative welfare impacts of each mitigation approach, it therefore

makes sense to focus on cases in which the addict and the chipper differ in their welfare

rankings of the three mitigation approaches. For this reason, we presume that the size of

the decreases in θ, λ, and π from harm-, cue-, and cravings-reduction, respectively, are

sufficiently “comparable” (i.e. neither too large nor too small in relation to each other)

to ensure that the addict and the chipper will not agree that any one mitigation approach

is preferable to another.

Proposition 7 Given the mitigation approaches are comparable (in the sense described

above), the addict’s welfare-rankings (from best to worst) are (1) cravings-reduction, (2)

harm-reduction, (3) cue-reduction. The chipper’s welfare rankings are reversed.

Proposition 7’s implication that addicts disproportionately benefit from cravings-reduction

fits with the apparent tailoring of nicotine replacement therapies to heavy smokers.17 In

turn, the implication that chippers (and nonusers, from Lemma 2) are best served by

cue-reduction fits with the real-world targeting of many cue-reduction policies towards

younger individuals — who are not only less likely to smoke, but are also significantly

17 While I am unaware of any studies that compare usage rates between addicts and chippers, thisimplicit view is revealed by the capacity of nicotine replacement treatments to sustain blood nicotinelevels on par with those observed in heavy smokers (Stead et al., 2008) as well as the standard exclusionof chippers from research on the efficacy of such products (see Okuyemi et al., 2002).

15

less likely (in comparison to adults) to smoke on a daily basis even if they do smoke.18

For instance, the 1997 Canadian Tobacco Act prohibited advertising tobacco products in

a way that can be “construed on reasonable grounds to be appealing to young people,”

while also forbidding print advertising — except in publications with at least 85 percent

adult readership. The 2009 U.S. Tobacco Control Act similarly introduced a host of

youth-oriented restrictions on marketing, such as a regulation on the use of colors and

graphics in tobacco advertisements, which exempted advertisements in adult magazines

or in adult-only retail establishments.

Building on this idea, the 2014 U.S. Surgeon General’s report proposed a different type

of regulation intended to reduce youth exposure to cues: requiring R-ratings on movies

depicting the act of smoking. Besides reinforcing the notion of targeting cue-reduction

policies towards those who lack established habits, the proposal also serves as a reminder

that consumption itself can function as a cue to others who view it — whether in movies

(Shmueli et al., 2010) or in real life (Ellickson et al., 2003). By this logic, policies that

restrict public consumption would reduce environmental cues as well.

4.3 Public Consumption Bans

To account for the notion that others’ consumption can serve as a cue, we now consider a

public consumption ban that imposes c = 0 while simultaneously reducing the cue-arrival

rate λ.

Proposition 8 There exist a s′ with 0 < s′ < 1 such that a public consumption ban

increases welfare for s < s′ and decreases welfare for s > s′.

Thus, public consumption bans benefit those with weaker habits and nonusers while hurt-

ing those with stronger habits. This prediction fits with previously-cited evidence that

such restrictions tend to be opposed by heavy smokers but supported by light smokers and

nonsmokers. The result is also broadly consistent with Philpot et al.’s (1999) findings on

behavioral responses to a nightclub smoking ban, as the ban leads to a disproportionately

large decrease in patronage among daily relative to nondaily smokers.

Furthermore, since s would decrease while such a ban is in effect, Proposition 8 implies

that someone who initially opposes a public consumption ban can come to support it after

its enactment (i.e., they would vote against repealing the ban). This aspect is in line with

evidence that, 8-9 months after the implementation of a comprehensive public smoking

ban in Ireland, there was greater public support for bans on smoking in workplaces (67%

vs. 43%), restaurants (77% vs. 45%), and bars/pubs (46% vs. 13%) than one year prior

18 Roughly 80 percent of smokers under 18 years old, yet only 40 percent of the overall US smokingpopulation are nondaily smokers (Substance Abuse and Mental Health Services Administration, 2014).

16

(Fong et al., 2006). A study from Norway similarly found an increase in support for

bans on smoking in bars and restaurants, from 47% six months before the ban to 58%

six months after enactment (Directorate for Health and Social Affairs, 2005).

The ET model’s implications for public consumption bans differ from those offered by

prevailing theories. The Laibson and Bernheim-Rangel cue theories — which also formal-

ize the notion that consumption restrictions would come hand-in-hand with reductions

in environmental cues — instead predict that addicts would benefit the most from such

policies, while those lacking any habit would not benefit. The rational addiction and

Gul-Pesendorfer temptation theories do not include environmental cues, thus preclud-

ing the notion that a public consumption ban would reduce their prevalence. Without

this (or any other) externality of consumption, these theories do not distinguish public

consumption bans from private consumption bans, of which only the Gul-Pesendorfer

temptation theory allows a strictly positive welfare benefit to nonusers — though, in

contrast to Proposition 8, those with strong habits would benefit even more.

5 Conclusions

This paper developed a theory of harmful addiction based on the so-called “stubbornness”

and “reinforcement” of unwanted urges. The theory captures a range of empirical pat-

terns that prevailing theories do not predict, such as concurrent increases in the frequency

and levels of consumption during habituation, short-term substitution in demand across

time (along with long-term complementarity), a relatively weak dependence of consump-

tion on environmental cues among addicts, and the emergence of relatively consistent

consumption patterns as a weaker habit develops into an addiction. In addition, the

theory offers new implications for restrictions on consumption and on advertising, and

also for individually-pursued mitigation options, such as harm-reduction (e.g. switching

from cigarettes to e-cigarettes) and cravings-reduction (e.g. wearing a nicotine patch).

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A Derivation of the Euler Equation

To derive the Euler equation (6), first substitute out V ′(si+1) in (5) using (4) with c = ci

and s = si, so that (1−σ)s+σc = si+1. Second, in the new expression for (5), iterate every

term forward one decision. Again, substitute out V ′(si+1) (after the iteration) using (4).

Third, substitute out V (si+2) using (3) iterated forward one decision (i.e. with c = ci+1

and s = si+1, so that (1 − σ)s + σc = si+2). After solving for V (si+1), insert the expression

and its once-iterated variant for V (si+2) into the left- and right-sides, respectively, of (3)

with c = ci+1 and s = si+1. Rearranging yields (6).

21

B Proofs

B.1 Proof of Proposition 1

Let θ(s ∣σ, γ) ≡ (1−σ)(γ+s)1+(γ+s)(1+σ(γ+1)−(1−2σ)s) , so that the steady-state equation can be expressed

as θ(s) = θ. We first consider the candidate steady-states sL and sH , respectively defined

as the smaller and larger roots to the quadratic implied by θ(s) = θ (closed-form solutions

for sL, sH , θ, θ, σ, and σ are provided in Appendix C). Using (9) and our definition of

θ, we can see: θ = θ implies sH = 12 and 0 < sL < sH ; and θ = θ implies sL = 0. Using

the implicit function theorem, we have ds(θ ∣θ(s)=θ)dθ =

((γ+s)((γ+1)σ+s(2σ−1)+1)+1)2(1−σ)((1−2σ)(γ+s)2+1) . Since the

numerator is positive, ds(θ ∣θ(s)=θ)dθ must have the same sign as (1− 2σ)(γ + s)2 + 1. Solving

for σ using θ(sH) = θ(sL), plugging the solution into (1−2σ)(γ+s)2+1, and rearranging,

we see ds(θ)dθ > 0 if and only if (γ+s)2

(γ+sL)(γ+sH) < 1, which clearly holds for s = sL but not for

s = sH . It follows that sL ∈ (0, 12) and sH > 1

2 for all θ ∈ (θ, θ). Given θ = θ, sH = 1 if σ =

σ, sH = 12 if σ = σ, and sH =

1−γ2(2σ−1)γ(2σ−1) for general σ. Differentiating the last term with

respect to σ gives − 2γ(2σ−1)2 < 0. Therefore, sH ∈ (1

2 ,1) for all σ ∈ [σ,σ] given θ = θ. SincedsH(θ)dθ < 0 and sH = 1

2 at θ = θ, sH ∈ (12 ,1) for all σ ∈ (σ,σ), θ ∈ (θ, θ).

Observing θ − θ ∝ 2 − (2σ − 1)γ(2γ + 1), θ > θ if and only if 2 − (2σ − 1)γ(2γ + 1) >

0. Since this expression is decreasing in σ and 2 − (2σ − 1)γ(2γ + 1) = 0, θ > θ at all

σ ∈ (σ,σ). In turn, σ − σ = (2γ(γ + 1)(2γ + 1))−1 > 0, as desired. By inspection, σ > 0

and θ > 0. Next, we see σ = 56 for γ = 1 and dσ

dγ = 4(2γ + 1)−2 − γ−2 < 4(2γ)−2 − γ−2 = 0,

implying σ < 1 for all γ so that [σ,σ] ⊂ [0,1].

By inserting (7) and (8) into (3) and evaluating at c = s, we see, for s ∈ {sL, sH},

V (s) = −(1 + θs)(γ + 1 + s). Using this expression, while applying the same procedure

to the Envelope condition, we see, for s ∈ {sL, sH}, V ′(s) = −(1+θs)(γ+1+s)

1+σ(γ+s) . Inserting

these expressions (with (7) and (8)) into (4), setting c = s, taking the (full) deriva-

tive with respect to s, substituting in θ(s) for θ, and then multiplying through by(γ+1+s)(1+σ(γ+s))(1+(γ+s)(1−s+σ(1+2s+γ)))

(1−σ) >0, we see the first-order condition is increasing in

s (maintaining that it is evaluated at c = s) from a steady-state if (1−2σ)(γ + s)2 +1 > 0.

As we already saw, this condition holds for s = sL but not for s = sH . This meansdc∗(sL)ds > 1 and dc∗(sH)

ds < 1. Since dc∗(sL)ds > 1, d[si+1−si]

dsi= σ(dc

∗(si−1)ds − 1) > 0 for si = sL,

implying sL is unstable. To establish the stability of sH , we draw from part (i) of Propo-

sition 2 (which does not rely on this Proposition and is proven in Appendix B.2), which

establishes dc∗(s)ds > 0. Along with dc∗(sH)

ds ∈ (0,1), and (1), dc∗(s)ds > 0 implies that, in a

neighborhood around sH , the {si} sequence strictly increases (decreases) below (above)

sH with sH serving as an upper (lower) bound. Therefore, {si} (and its accompanying

{ci} sequence) converges to some limit, call it s. Since s is a steady-state and the steady-

22

state condition (9) is quadratic, there are no solutions to θ(s) = θ besides sL and sH . This

along with the fact that s is in a sufficiently small neighborhood near sH means s = sH , so

that si converges to sH , a stable steady-state. The same logic extends to any si ∈ (sL,1]

implying {si} converges to sH for all s0 ∈ (sL,1]. Analogously, since {si} is decreasing

on si ∈ (0, sL) and is bounded below by s = 0, {si} converges to 0 for s0 ∈ [0, sL). Since

sL = 0 for θ = θ, c∗(0 ∣θ > θ) ≤ c∗(0 ∣θ = θ) = 0. With our previous work, this establishes

s = 0 as a stable steady-state. ∎

B.2 Proof of Proposition 2

If the Bellman’s maximization argument, Λ(s, c) ≡ −(1 + θc) +D(s, c)V (s+) with s+ =

(1 − σ)s + σc, is supermodular, i.e. satisfying the increasing differences property, it

follows from Topkis’ Theorem that c∗(s) is increasing in s. Hence, a sufficient con-

dition for this is Λsc(s, c) > 0. Using (4) to substitute V ′(s+) out of Λs(s, c) = (1 −

σ)D(s, c)V ′(s+) +Ds(s, c)V (s+) and plugging in our expressions for D(s, c) and Ds(s, c),

we see Λs(s, c) =(1−σ)θσ +

(γ+σ+s+)V (s+)σ(γ+1+c)2 . We can now calculate Λsc(s, c) =

(γ+σ+s+)V ′(s+)(γ+1+c)2 −

((2−σ)(γ+s)+σ(1+c))V (s+)σ(γ+1+c)3 . Substituting out V ′(s+) using (4), we get Λsc(s, c) =

(γ+σ+s+)θσ(γ+s)(γ+1+c)

−(1−σ)(γ+s)V (s+)

σ(γ+1+c)3 . Since both terms in this expression are positive (adding a positive and

subtracting a negative), Λsc(s, c) > 0, as desired.

To show τ(s, c∗(s)) decreases in s, let Λ(s,D) ≡ Λ(s, c(s,D)), where c(s,D) ≡ {c ∶

D(s, c) =γ+sγ+1+c = D} =

γ+sD − γ − 1. Differentiating, we see Λs=Λs+Λccs (arguments

of Λ, Λ, c, and their partial derivatives are suppressed). Differentiating again, this

time with respect to D, we get ΛsD=Λscc+ΛccsD+ΛcccscD. Observing the first-order

condition Λc(s, c(s,D)) = 0 and substituting in cs(s,D) = D−1 and cD(s,D) = −γ+sD2 ,

we see ΛsD=−γ+sD2 (Λsc +

ΛccD

). Using (5) to substitute V ′(s+) out of Λc(s, c) = −θ +

σD(s, c)V ′(s+) + Dc(s, c)V (s+), we see Λc(s, c)=(γ+σ+s+)V (s+)(1−σ)(γ+1+c)2 +

σV ′(s)1−σ −θ. Differentiat-

ing with respect to c again and using (4) to substitute out V ′(s+), we get Λcc(s, c) =(γ+s)V (s+)(γ+1+c)3 −

θ(γ+σ+s+)(1−σ)(γ+s)(γ+1+c)2 . Inserting this expression along with our previous expression

for Λsc(s, c) into ΛsD=−γ+sD2 (Λsc+

ΛccD

), substituting out D (and partial derivatives) for the

corresponding expressions in terms of s and c, and simplifying gives ΛsD =θ(γ+1+c)3(γ+σ+s+)

γ+s− (1−σ)(γ+1+c)(γ+s)V (s+). Since the first and second terms are both positive, ΛsD > 0.

Therefore, from Topkis’ Theorem, D(s, c∗(s)) is increasing in s. Since D(s, c) = e−τ(s,c),

τ(s, c∗(s)) is decreasing in s. ∎

23

B.3 Proof of Proposition 3

Noting N `(c ∣s) ≡ max{n ∶ ti+n ≤ ti + `, ci = c, si = s}, observe that dti+1dci

= τc(si, ci) =1

γ+1+ci > 0. Since c∗(s) is increasing in s (Proposition 2), if si+n is increasing in ci, then

si+n+1 = (1 − σ)si+n + σc∗(si+n) is too. Thus, given si+1 is increasing in ci from (1), si+k is

increasing in ci for all k ≥ 1. Since τ(s, c∗(s)) is decreasing in s (Proposition 2), it follows

that dti+1+kdci

−dti+kdci

=dτ(si+k,c∗(si+k))

dci< 0. Hence, dti+1+k

dcidecreases in k. Let tni (ci, . . . , ci+n ∣si) ≡

{ti+n ∶ ci, . . . , ci+n, si}. Noting∂t∞i∂ci+k

=∂t∞i+k∂ci+k

,∂t∞i∂ci

> 0 for all ci, ci+1, . . . and si implies∂t∞i∂ci+k

> 0.

Since ci+k is increasing in ci for k > 0,∂t∞i∂ci

> 0 implies dt∞dci

> 0. Next, observe that∂t∞i∂ci

=

τc(si, ci)+σ∑∞j=1(1−σ)

j−1τs(si+j, ci+j)=1

γ+1+ci − σ∑∞j=1(1−σ)

j−1 1γ+si+j <

1γ+1+ci −

σ∑∞j=1(1−σ)j−1γ+maxj>0{si+j}

= 1γ+1+ci −

1γ+maxj>0{si+j} . Since 1 + ci > maxj>0{si+j}, dt∞

dci< 0. Since dti+1

dci> 0 and dti+1+k

dci

decreases in k, this gives the desired result. ∎

B.4 Proof of Lemma 1

With cues, V (si) = maxci

{−(1 + θci) +E[e−(ti+1−ti)]V (si+1)}. Comparing this to the deter-

ministic Bellman with D0(s, c) = e−τ0(s,c), V (si) = max

ci{−(1+θci)+D0(si, ci)V (si+1)}, we

see the problems are equivalent if and only if D0(si, ci) = E[e−(ti+1−ti)]. With Pr[ti+1− ti =

τ(si, ci)] = D(si, ci)λ, E[e−(ti+1−ti)] = ∫τ(si,ci)

0 λe−(λ+1)tdt + Pr[ti+1 − ti = τ(si, ci)]D(si, ci)

= (1 + λ)−1(λ + (γ+siγ+1+ci )

1+λ) = D0(si, ci) so that τ 0(s, c) = − ln(D0(s, c)) = ln(1 + λ) −

ln(λ + (γ+sγ+1+c)

1+λ). D0(s, c) can then be inserted into (6) along with u(c) = −(1 + θc), to

obtain an expression describing the optimal consumption path with λ ≥ 0. We can then

set each c = s to obtain the corresponding steady-state condition. Applying the implicit

function theorem, we can see that (and how) the value of s satisfying the steady-state

condition varies continuously with λ. Given sL ∈ (0, 12) and sH ∈ (1

2 ,1) for λ = 0, it

follows that sL ∈ (0, 12) and sH ∈ (1

2 ,1) must still hold for sufficiently small λ > 0. Thus

the chipper and addict steady-states will continue to exist. The sign of the first-order

condition evaluated at c = s = 0 will likewise be robust to sufficiently small departures

from λ = 0, implying c∗(0) = 0. ∎

B.5 Proof of Proposition 4

The probability consumption at ti+1 coincides with a cue is Pr[ti+1 − ti < τ(si, c∗(si))] =

1−e−λτ(si,c∗(si)), which is increasing in τ(si, c∗(si)), and thus decreasing in si (equivalently,

in si+1) as λ→ 0 given Proposition 2. ∎

24

B.6 Proof of Proposition 5

Suppressing arguments of τ(si, c∗(si)), Var[ti+1−ti] = ∫τ

0 t2λe−λtdt+τ 2e−λτ − [∫

τ

0 tλe−λtdt+

τe−λ]2 = λ−2(1−e−2λτ −2λτe−λτ). ∂Var[ti+1−ti]∂τ = 2λ−1e−2λτ(e−λτ + λτ − 1) > 0 for all τ > 0.

Since τ(si, c∗(si)) is decreasing in si (Proposition 2), Var[ti+1 − ti] is too. ∎

B.7 Proof of Proposition 6

By the optimality of c∗(s), permanently fixing c = 0 is strictly welfare-reducing if c∗((1−

σ)ns) > 0 for any n = 0,1, . . ., and welfare-neutral if c∗((1 − σ)ns) = 0 for all n. Since

(1−σ)ns decreases in n and c∗(s) increases in s (Proposition 2), c∗((1−σ)ns) > 0 for any

n is equivalent to c∗(s) > 0, and c∗((1 − σ)ns) = 0 for all n is equivalent to c∗(s) = 0. ∎

B.8 Proof of Lemma 2

For any {ci}∞i=0, it is straightforward to show that expected lifetime utility, −∑i e−E[ti](1+

θci), is weakly decreasing in θ, in λ (noting E[ti] decreases in λ, ceteris paribus), and

in π given τ(s, c) = (1 − π) ln(γ+1+cγ

) + π ln(γ+1+cγ+s ). Therefore reducing θ, λ, or π weakly

improves welfare. From u(c) = −(1+θc), it is clear that harm-reduction is welfare-neutral

if and only if c∗(s) = 0. From τ(s, c) = (1 − π) ln(γ+1+cγ

) + π ln(γ+1+cγ+s ), cravings-reduction

is welfare-neutral if and only if s = 0. Given τ(s, c∗(s)) is decreasing in λ for all s, c,

cue-reduction is strictly welfare-improving for all s ∈ [0,1]. ∎

B.9 Proof of Proposition 7

Let s ∈ {sL, sH}. Using (3), we can see from our given values θ, λ = 0, and π = 1,

Vθ(s) = −s(γ + 1 + s), Vλ(s) = −(1 + θs)(γ + 1 + s)(1 − (γ + s) ln(γ+1+sγ+s )), and Vπ(s) =

−(γ + s)(γ + 1 + s)(1 + θs) ln(γ+sγ ). Since λ ≥ 0 is sufficiently small and bounded below

by zero, the decrease in λ from cue-reduction is also sufficiently small. Given this, the

fact that Vλ(s), Vθ(s), Vπ(s) ∈ (−∞,0), and the property that the welfare gain from

cue-reduction is higher at either sL or sH when compared to each among harm- and

cravings-reduction, the decreases in θ and π can both be taken to be sufficiently small.

Consequently, we can work with Vθ(s), Vλ(s), and Vπ(s) evaluated at the original θ, at

λ = 0, and at π = 1

For each a, a′ ∈ {θ, λ, π} with a ≠ a′, we know either (i) Va(sH) > Va(sL) and Va′(sH)

< Va′(sL), or (ii) Va(sH) < Va(sL) and Va′(sH) > Va′(sL). Since Va(s) < 0 for each

a ∈ {θ, λ, π} (Lemma 2), case (i) must apply if Va(s)Va′(s)

is increasing in s, while case (ii)

must apply if Va(s)Va′(s)

is decreasing in s. Thus, harm-reduction is better than cue-reduction

25

at sH but not at at sL if Vλ(s)Vθ(s) = (1

s + θ)(1 − (γ + s) ln(γ+1+sγ+s )) > 0 is decreasing in s.

Since 1s + θ is decreasing in s, it suffices to show d

dx[x ln(x+1

x)] = ln(x+1

x) − 1

1+x > 0, where

x = γ + s. Since ln(x+1x

) − 11+x = 0 given x = ∞ and d2

dx2[x ln(x+1

x)]=− 1

x(1+x) < 0 for all

x > 0, ddx

[x ln(x+1x

)] > 0 for all x ∈ (0,∞), implying Vλ(s)Vθ(s) is decreasing in s, as desired.

Similarly, cravings-reduction is better than harm-reduction at sH but not at sL if Vπ(s)Vθ(s) =

s−1(1+θs)(γ+s) ln(γ+sγ ) is increasing in s. Differentiating by s then multiplying through

by s > 0, the desired condition becomes s(1 + θs) − (γ − θs2) ln(γ+sγ ) > 0. This expression

equals zero for s = 0 and its derivative for general s is (γ+θs)sγ+s + 2θs ln(γ+sγ ) > 0, implying

the inequality holds for all s > 0. ∎

B.10 Proof of Proposition 8

From the proof of Proposition 2, the Bellman’s maximization argument Λ(s, c) is super-

modular. Therefore, k > 0 and k′ > 0 imply Λ(s+k, c+k′)−Λ(s+k, c) > Λ(s, c+k′)−Λ(s, c).

Taking k′ = c∗(s) and c = 0, we see Λ(s, c∗(s)) − Λ(s,0) < Λ(s + k, c∗(s)) − Λ(s + k,0)

< Λ(s + k, c∗(s + k)) − Λ(s + k,0), where the last inequality follows from the optimal-

ity of c∗(s + k) given habit stock s + k. Thus, a temporary (one-time) ban hurts more

at s + k than at s. Now suppose a ban will be in effect for n urges starting from the

present. Then the choice of c when the ban is no longer in effect can be expressed as

V (s) = maxc{−(1+∑n−1i=0 ∏

ij=0D((1−σ)js,0))+∏

n−1j=0 D((1−σ)j,0)Λ((1−σ)n−1, c)}. Since

Ds(s,0) > 0, the welfare cost of extending the temporary ban to include the next urge is

increasing (in magnitude) with s. Given this property, and the fact that the (expected)

welfare benefit of decreasing the cue-arrival rate from λ to λ′ < λ increases continuously

with λ, holding λ′ fixed, in the stochastic setting, we can be assured that taking λ > 0 to

be sufficiently small implies public consumption bans necessarily reduce welfare for large

enough s. Now a public consumption ban strictly improves welfare for sufficiently small

s, as c∗(s) = 0 for sufficiently small s, in which case the cost of the ban is zero while

the benefit from decreasing λ is nonzero. Given these properties, it follows that some

threshold s′ ∈ (0,1) exists for which the public consumption ban is welfare-reducing for

s > s′ and welfare-improving for s < s′. ∎

C Closed-Form Expressions

This section provides closed-form solutions for relevant values referenced in the results.

To start, we use (9) to derive the interior steady-states in Proposition 1:

sL = (δ +√δ2 + 1 − 2σ)−1 − γ, sH = (δ −

√δ2 + 1 − 2σ)−1 − γ,

26

where δ ≡ 1−σ2 (1 + θ−1 − γ) − 1. Since c∗(s) = s in a steady-state, we know sL and sH

define the steady-state consumption levels. Using (8), we can derive the corresponding

time-intervals between consumption occasions:

τ(sL, sL) = ln(1 + δ +√δ2 + 1 − 2σ), τ(sH , sH) = ln(1 + δ −

√δ2 + 1 − 2σ)

The boundaries on θ and σ given in Proposition 1 are

θ =(1 − σ)γ

1 + γ(1 + σ(γ + 1)), θ =

2(1 − σ)(2γ + 1)

1 + 2(γ + 2)(1 + σ(2γ + 1)), σ =

γ2+ γ + 1

2γ(γ + 1), σ =

2γ2+ γ + 2

2γ(2γ + 1).

Note, since θ and θ depend on σ and all of these bounds depend on γ, the restrictions

θ ∈ (θ, θ) and σ ∈ [σ,σ] collectively represent a joint condition on θ, σ, and γ.

D A Behaviorally-Equivalent Utility Formulation

In the ET model, the utility function can be expressed as u(c) = −1+f(c), where f(0) = 0

and f ′(c) < 0 for all c ∈ [0,1]. (Under our functional form assumption in (7), f(c) = −θc.)

This appendix considers an alternate utility function given by

u(s, c) = f(c) −D(s, c),

where D(s, c) = e−τ(s,c). As defined here, u(s, c) can be understood as a modification

of the original utility function that includes the cost of the next urge (in present value)

instead of the current −1 urge cost. This shifting of each urge cost to the utility function

at the previous urge is in essence a notational relabeling and it is readily verifiable that

the dynamic optimization problem is not affected.19

D.1 Reinterpreting u(s, c)

Given the behavioral-equivalence of the models with u(c) and u(s, c), we can re-interpret

u(s, c) in other ways. In particular, instead of interpreting the −D(s, c) term in u(s, c) =

f(c)−D(s, c) as the present value of the next urge cost and f(c) as the direct (dis)utility

from consumption, u(s,0) = −D(s,0) could be interpreted as the current urge cost and

f(s, c) ≡ u(s, c) − u(s,0) = f(c) −D(s, c) +D(s,0) as the direct (dis)utility from con-

sumption.

19 That is, compared to the original Bellman equation V (s) = maxc{u(c) +D(s, c)V ((1 − σ)s + σc)},the Bellman corresponding to u(s, c) simply moves the discounted cost of the next urge, −D(s, c), fromthe future value function to the present utility function while removing the sunk present -1 urge costfrom the optimization argument, neither of which affect the optimal consumption choice c∗(s).

27

One noteworthy feature of u(s, c) under this re-interpretation is that, given our rein-

forcement assumption τs(s, c) < 0, the effective urge cost u(s,0) = −D(s,0) decreases in

s (thus increasing in magnitude) as seen from us(s,0) = −Ds(s,0) = τs(s,0)e−τ(s,0) < 0.

Thus, while the urge cost under u(c) is fixed at −1, u(s, c) can accommodate the realistic

notion that an urge might be more costly if experienced at higher s. For example, a bio-

logical craving could very well be more costly (whether hedonically or in its attentional

demands) than a milder urge arising in the absence of a prior habit.

Another noteworthy feature of u(s, c) is that, unlike u(c), u(s, c) could increase with c

since uc(s, c)=f ′(c) −Dc(s, c) with f ′(c) < 0 yet −Dc(s, c) = τc(s, c)e−τ(s,c) > 0. Using our

functional form assumptions (7) and (8), this effective marginal utility of consumption is

uc(s, c) = −θ +γ + s

(γ + 1 + c)2.

Also observe that usc(s, c)=−Dsc(s, c)=γ

(γ + 1 + c)2>0, revealing (under our re-interpretation

of u(s, c)) a marginal utility of consumption that grows with past consumption. Even

with usc(s, c) > 0, however, we can verify the effective marginal utility of consumption

from c = 0 and at s = 0, i.e. uc(0,0), is still positive given our parametric restrictions

θ ∈ (θ, θ) and σ ∈ [σ,σ].20 Thus, despite the assumption of u′(c) < 0, the ET model

under u(s, c) can accommodate a notion that consumption brings positive utility even

for someone who lacks a prior habit. In this case, the motive for consumption can be

understood as a composite of the incentive from uc(s, c) > 0 and from τc(s, c) > 0. We

note here that with the next urge cost (as originally interpreted) −D(s, c) now embedded

in u(s, c) and interpreted as part of the present utility, the stubbornness assumption

τc(s, c) > 0 still creates an incentive to consume as a means to delay the next urge. The

difference is that the next urge cost (in present value) is now given by D(s, c)u((1 −

σ)s + σc,0)=−D(s, c)D((1 − σ)s + σc,0) instead of −D(s, c), implying this incentive is

diminished relative to the original formulation with u(c) (since D((1 − σ)s + σc,0) < 1).

D.2 Relationship to Rational Addiction Theory

The defining feature of the habit-formation preferences of Becker and Murphy’s (1988)

standard rational addiction theory is that the marginal utility of consumption increases

with past consumption. With usc(s, c) > 0, the basic ET model (without cues) could

therefore be interpreted as an integration of rational addiction-style preferences through

20 Note, uc(0,0)= −θ +γ

(γ+1)2is positive as long as γ

(γ+1)2>θ. Using our definitions of σ and θ from

Appendix C, we can see σ=arg maxσ∈[σ,σ]{θ(σ, γ)}, implying γ(γ+1)2

>(2γ+1)(γ2

+γ−1)2γ4+9γ3+16γ2+12γ+2

is a sufficient

condition for uc(0,0)>0. Multiplying through by the denominators and rearranging, we can see thecondition is equivalent to 2γ4 + 9γ3 + 12γ2 + 5γ + 1 > 0, which clearly holds.

28

u(s, c) and endogenously-timed urges through τ(s, c). With that said, the equivalence

between the model under u(s, c) and under u(c) demonstrates that such effective habit-

formation preferences are not needed to generate the ET model’s unique behavioral

predictions, as the same patterns can be captured by a model in which s only influences

behavior through τ (as opposed to influencing behavior through both τ and u).

E Classifying Other Theories’ Predictions

This section explains how other theories’ predictions (as listed in Table 1) were classified.

We first explain some implicit restrictions and definitions that were used to clarify the

exposition, and then describe item-by-item the basis for each prediction.

E.1 Implicit Restrictions

To avoid potential sources of ambiguity, in classifying each theory’s predictions we pre-

sume model parameters are stable and abstract from pre-commitment opportunities that

are considered in some models.21 In addition, to allow a common standard for compar-

ison (as discussed in footnote 3), we presume that the “strength” of a habit is defined

by the associated state variable representing a time-weighted stock of past consump-

tion (analogous to s in the ET model).22 Under the conventional presumption that an

individual does not begin in a state of or exceeding addiction (in the ET model, this

would mean s0 < sH), statements describing how consumption behavior varies between

stronger and weaker habits can equivalently be understood as describing how behavior

varies between older and newer habits (respectively), all else equal. In the ET model, the

(simultaneous) increases in the habit stock and in the duration (i.e. “age”) of the habit

are also accompanied by increases in the consumption frequency and in the per-occasion

consumption levels (Proposition 2). For this reason, we can equivalently understand the

ET model’s predictions that invoke habit strength in terms of any of these four candidate

measures — habit stock, habit duration, consumption frequency, consumption levels —

21 Most notably, pre-commitment can lead to abstinence among addicts. For example, a drug addictmay check into a rehabilitation clinic that prevents future consumption. In this way, addiction couldtechnically entail zero consumption — a property that may seem rather paradoxical out of context, whileproviding little insight into how observable consumption behavior varies with habit strength.

22 In Becker-Murphy rational addiction theory, the habit stock is modeled analogously, except itevolves continuously over time instead of at discrete times. In Laibson cue theory, there are two suchstate variables — one corresponding to each “color” of cue that may arise. If the individual consumesfor “green” but not “red” cues, for example, the state variable associated with “green” consumption isthen naturally taken as the effective measure of habit strength. In Gul-Pesendorfer temptation theory,the relevant state variable is simply the previous consumption choice (as it would be in the ET modelif σ = 1). In Bernheim-Rangel cue theory, the state variable increases/decreases by 1 after a periodwith/without consumption (with a floor and ceiling).

29

though each of the latter two measures cannot provide a common standard for classify-

ing the predictions of other theories since, for each measure, some theories do not permit

variation in that measure.

Lastly, since the Bernheim-Rangel cue theory features a particularly high degree of

parametric and behavioral flexibility, we restrict our consideration of this theory to spec-

ifications that adhere to their motivating concept of addiction as featuring stochastic

cue-induced “mistakes.” This restriction is also implicit in the authors’ characterization

of their model in relation to others. For example, when discussing other theories of ad-

diction, Bernheim and Rangel (2004) write “while all of these theories contribute to our

understanding of addiction ... none of these models depicts addiction as a progressive

susceptibility to stochastic cues that can trigger mistaken usage.”

E.2 Item-By-Item Explanations

ET RA GP-T L-C BR-C

(I)how consumption ‘amounts’ changeas weaker habit turns into addiction

frequency

levels

↑

↑

0

↑

0

↑

0

0

↓

0

In Becker and Murphy’s (1988) rational addiction theory (RA), which is cast in con-

tinuous time, the flow consumption variable is always positive and increasing with the

habit stock on the path to the addicted steady-state. Since consumption occurs contin-

uously, the growth of the habit does not affect how often consumption occurs.23 In Gul

and Pesendorfer’s (2007) temptation theory (GP-T), consumption levels increase mono-

tonically with past consumption, with consumption occurring once in each discrete-time

period. In Laibson’s (2001) cue theory (L-C), a steady-state of nonzero consumption

entails a consumption level of c = 1 with some probability p ∈ (0,1] in each discrete-time

period, while consumption does not occur (c = 0) with probability 1 − p. On the path to

this steady-state, the probabilities of c = 1 and of c = 0 occurring in each period are the

same as in the steady-state, thus implying no changes in the frequency or the levels of

consumption. Bernheim and Rangel’s (2004) cue theory (BR-C) also features a binary

consumption choice so that only one nonzero consumption level (c = 1) is possible. In

turn, addiction entails stochastic cue-induced mistakes (implying consumption is proba-

bilistic), but it is preceded by a period of “casual use” in which consumption occurs in

every period, thus implying a higher consumption frequency before becoming addicted.

23 While the present exercise takes other theories’ predictions as is, one could argue that an increasingflow rate of consumption is a stylized representation that would more naturally be interpreted as anincreasing frequency of consumption. Even under this interpretation, Becker-Murphy rational addictiontheory does not simultaneously capture the increases in the levels and frequency of consumption.

30

ET RA GP-T L-C BR-C

(II)present demand & future demand:(C)omplements or (S)ubstitutes?

short-run

long-run

S

C

C

C

C

C

C

C

C/S

C/S

In rational addiction theory, future demand (which would naturally be measured by

cumulative consumption during the associated time period), unambiguously rises with

present demand regardless of the time-horizon, thus implying both short- and long-run

complementarity. This is also the case in Gul-Pesendorfer temptation theory. In the

Laibson cue theory, there is a threshold habit strength (specific to the “color” of the

present cue) such that present and future consumption levels whenever a cue of that

color arises will be c = 0 below the threshold and c = 1 above the threshold. As a

consequence, there is a region near this threshold for which all future consumption levels

when that color of cue arises will be determined by — and the same as — the present

consumption choice, c ∈ {0,1}, thus capturing both short- and long-run complementarity

(albeit weakly, in that future consumption is not affected by the present consumption

level if the habit stock specific to that color is not sufficiently close to the threshold

between c = 0 and c = 1).24 The Bernheim-Rangel cue theory does not offer specific

predictions concerning the relationship between present and future demand. At s = 0 for

instance (as defined in their model), an idiosyncratic choice of c = 1 could lead to higher

demand — thus implying complementarity — in the short- and long-run by bringing the

decision-maker to the casual user phase in which consumption will continue. If freshly

addicted, however, an idiosyncratic choice of c = 0 can lead to higher (expected) demand

in the next period by causing the individual to return to the “intentional use interval” in

which consumption always occurs (one state higher, however, and the opposite may be

true). For longer time-horizons, the direction of the effect is ambiguous, and may change

multiple times as the time-horizon is lengthened one period at a time.

ET RA GP-T L-C BR-C

(III)how dependence of consumption on environmentalcues changes as weaker habit turns into addiction

↓ 0 0 0 ↑

Since the rational addiction and Gul-Pesendorfer temptation theories do not feature24 This form of intertemporal complementarity aligns with the measures used in Proposition 3 as it

suggests the number of future consumption occasions increases with current demand over any time-horizon. The rational addiction and Gul-Pesendorfer temptation theories also permit analogous predic-tions whereby positive future consumption may only occur following positive present consumption neara threshold habit level. That said, the forms of complementarity (as first described for these theories)in which demand corresponds to aggregate consumption levels are instead directly comparable to thedemand measures considered in Proposition O2 of the online appendix, which re-establishes adjacentsubstitution and distant complementarity in the ET model using this alternate demand measure.

31

environmental cues, consumption is independent of environmental cues and therefore the

extent of the dependence does not vary on the path to addiction. In the Laibson cue

theory, consumption will likewise be independent of environmental cues if the per-period

probability of consuming in the steady-state of addiction is p = 1. If the addicted steady-

state instead features p < 1, consumption is entirely dependent on environmental cues

in that the decision-maker always consumes (c = 1) if a green cue is present and always

abstains (c = 0) if a red cue is present (or vice versa). In either case, the degree of the

dependence remains the same as a habit strengthens into an addiction. In Bernheim-

Rangel cue theory the dependence of consumption on cues increases on the path to

addiction in that consumption is independent of cues during the casual use phase, but

consumption becomes dependent on cues once addicted.

ET RA GP-T L-C BR-C

(IV)how variability of consumption schedules

changes as weaker habit turns into addiction↓ 0 0 0 ↑

Since the times at which consumption occurs in the rational addiction and Gul-

Pesendorfer temptation theories are deterministic, there is no variability (or changes

thereof) in consumption schedules. In the Laibson cue theory, the probability of con-

sumption in each period is fixed, implying the variability (or more precisely, the variance

of the time-interval between consecutive consumption occasions) does not vary with habit

strength on the equilibrium path. In the Bernheim-Rangel theory, consumption is de-

terministic during the casual use phase but becomes stochastic upon transitioning into

an addiction, implying an increase (from zero) in the variance of the time between con-

sumption occasions.

32