2. Rational Addiction Model. 2.1 Background ideas. Static consumer theory treats consumption choices as intertemporally independent - choices at one point of ...
Talking Rationally About Rational Addiction Harry Clarke and Svetlana Danilkina Department of Economics and Finance La Trobe University (revision March 19 2006)
Abstract. This paper reviews the rational addiction (RA) model of Becker and Murphy (1988) (BM) to demonstrate how its conclusions are based on assumptions about model parameters. The BM analysis emphasises the existence of an unstable steady state but the presumption of instability imposes strong restrictions on model parameters and is unlikely. Moreover, with two steady states (one stable, one unstable) the stable steady state may correspond to a lower level of addiction than the unstable one, the reverse of the BM finding. This implies that the effects of price changes on consumption, can be distinct from those given by BM. Also while the relative size of long-run and short-run effects depends on adjacent complementarity and addiction depreciation as BM suggest, short-run effects of a price change are generally greater than envisaged by BM. Other features of the RA model such as ‘cold turkey’ and ‘binge’ behaviour are reconsidered. The issue of detoxification is explicitly analysed here.
1. Introduction
Consumption of a substance is addictive if, as the duration and intensity of consumption increases: (i) users become more tolerant to the substance so they progressively want more of it; (ii) users find it increasingly difficult to cease consumption; (iii) if users do quit, they suffer withdrawal disutility. Finally, (iv), if use is interrupted or ceases, users experience unpleasant, and perhaps long-term, cravings for the substance.
Clearly, there are various behavioural motivations associated with addictive consumption. With increasing demand for an addictive activity, it is progressively more costly in the short-term for an agent to quit an addiction. There may also arise
2 longer-term cravings that reinstate addictive urges. This mix of motivations is part of the reason that treating of harmful addictions successfully has proven difficult.
2. Rational Addiction Model
2.1 Background ideas. Static consumer theory treats consumption choices as intertemporally independent - choices at one point of time are independent of those at other times. With effective capital markets, however, static budget constraints need to be replaced with an intertemporal constraint that allows borrowing and lending. Then consumption decisions become interdependent and can be optimally ‘smoothed’ over time to optimise the present value of intertemporal utility consistent with the intertemporal budget.
Consumption decisions are also intertemporally dependent when current consumption fashions future tastes. This occurs, with addictive goods, when intertemporal consumption utilities are optimised. Then current choices depend on past and expected future usage decisions even if effective capital markets are inoperative.
Becker and Murphy (1988) (BM), provide a comprehensive account of addiction that links economics and neuroscience in a ‘rational choice’ framework. Consumers of potentially addictive products optimise intertemporally-dependent preferences given time-consistent exponential discounting of future utilities, perfect foresight and are subject to an intertemporal budget constraint. It is supposed that “…addictions, even strong ones, are usually rational in the sense of involving forward-looking maximization and stable preferences…” (ibid, p. 675). At core this is a model of endogenous tastes where current consumption drives future consumption given ‘full prices’ reflecting market price and costs of use in terms of future addiction.
Since BM assumes constant exponential discounting, they avoid issues of time inconsistency and self-control. Comparing their approach with self-control models, they state ‘By contrast, in our model present and future consumption are complements, and a person becomes more addicted at present when he expects events to raise his future consumption. That is, in our model, both present and future behaviors are part of a consistent maximising plan’ (ibid, p. 691-692). BM does not
3 assume users derive positive utility from being addicted and, indeed, addiction can be caused by unhappiness with, drugs consumed as a form of self-medication. But such addicts would be less happy if prevented from consuming. The initial unhappiness may be due to a temporary event so, as BM note, temporary events may trigger a permanent addiction.
2.2 Formal model. In the BM model a representative consumer
T
maximises U(0) = ∫ e −σt u[y(t), c(t), s(t)]dt 0
s '(t) = c(t) − δs(t) − h[D(t)]
subject to
T
and
T
− rt − rt ∫ e [y(t) + pc (t)c(t) + pd (t)D(t)]dt ≤ A0 + ∫ e w(s(t))dt . 0
0
where
U(0) = present value of consumption utilities at some base time t = 0; u(.) = instantaneous utility; c(t) = consumption of an addictive good at t; y(t) = consumption of some numeraire good at t; s(t) = stock of addiction capital or the level of addiction at time t;
σ = constant rate of time preference; T = length of life of the individual;
δ = constant depreciation rate of the level of addiction; D(t) = expenditure (effort) toward reducing the addictive dependence; h = corresponding “endogenous depreciation”; r = constant interest rate; pc(t) = price of addictive consumption in terms of the numeraire good at t; pd(t) = price of effort in reducing addictive dependence in terms of the numeraire good; A0 = initial value of assets; w(s(t)) = wages at time t as a function of the agent’s degree of addiction.
4 Instantaneous utility u is a strongly concave function of y, c and s that depends positively on y and c and negatively on s with ucs > 0. This implies that, with increasing addiction, agents value the addictive good more at the margin which is a necessary condition for reinforcement so past consumption raises current consumption. It has the implication also that, if consumption ceases, the individual will experience greater disutility the more he/she is addicted.
That us < 0 implies that the higher current consumption reduces future utility by increasing addiction, so the addiction envisaged is harmful. However given the agent’s perfect foresight this disutility is internalised by the consumer so its prospect is not, in itself, a basis for public intervention to limit consumption. If such negative effects of addiction on current consumption utility were not internalised, they become
internalities which constitute a potential case for public intervention to limit consumption based on consumer ignorance.
The earnings function w(.) is assumed to be concave and decreasing in s so addiction is assumed to reduce earnings at an increasing rate.
For the most part in their formal analysis BM assume: (i) the agent lives forever, so T is infinite; (ii) the rate of time preference equals the discount rate (so r = σ) and (iii) expenditure to reduce the addictive stock is ignored (D = 0). Even with these strong assumptions the solution procedure adopted by BM is non-straightforward because, (i) the BM solution procedure uses complex shadow prices, and, (ii) because many BM conclusions relate to a nonlinear model not actually analysed by them.
The intuition of the BM is straightforward. An agent inherits a certain degree of addiction s(0) at time 0. This might be large if the consumer has suffered a traumatic experience, such a personal tragedy, or because they have an intrinsically addictive personality. The agent then makes intertemporal consumption choices with respect to addictive and non-addictive goods which account for forecast effects of usage at each time on future utilities, via addiction dynamics. They also account for their intertemporal budget. Different life experiences, as summarised by their initial degree of addiction, drive different initial levels of addictive consumption even for people with the same wealth and preferences.
5
2.3 Implications. Several possible consumption patterns are possible for a rational addict.
(i) Given strong concavity of u the agent will generally, although not invariably, consume at least some of the addictive good, at least initially. (ii) Ignoring actions that can reduce dependence (so D(t) = 0 for all t) preferences can be devised for which a consumer will select constant consumption of the addictive good c(t) = δs(0) for all time t. Then the consumer will consume the addictive good forever without ever increasing their addiction: Zinberg (1984) refers to controlled and sustained use of potentially addictive drugs, such as heroin, as ‘chipping’. The BM analysis is consistent with chipping with controlled consumption. (iii) One can also devise preferences for which optimal c(t) < δs(0) for all t so that a consumer, starting with a certain preference for an addictive good, would use it at a diminishing rate and asymptotically would phase out consumption. (iv) Agents can consume the addictive good at rates that yield high initial consumption utilities for a relatively short period and then, for large enough rates of time preference, low utilities for a much longer period. Indeed, rationalisation of a prolonged long-term stream of disutility following an initial phase of positive utility can always be made explicable by selecting a high enough rate of time preference. Thus, although BM supposes user foresight, addiction is rationalised by making agents attach sufficiently low weight to future utilities. (v) Finally, BM dichotomises addictive behaviour into stable and unstable optimal dynamic patterns of adjustment. We shall discuss this dichotomy at length.
2.4 Critique. Some of the BM conclusions are set out more completely in Becker, Grossman and Murphy (1991) (BGM 91) and Becker, Grossman and Murphy (1994) (BGM94). These papers also discuss empirical implications, with BGM94 providing a detailed discussion of cigarette addiction. The key empirical implication of the BM model is that past and future expected prices influence current consumption. In particular, the proposition that expected higher future prices result in lower consumption today as is implied by forward-looking behaviour is tested. empirical studies, and certainly those cited, support this hypothesis.
Most
6 However two criticisms can be made of such studies. First, they assume that individuals are forecasting prices in advance when there is usually no basis for doing so. For example, even cigarette prices are seldom pre-announced. Second, forwardlooking behaviour does not in itself imply time consistency, a key assumption of the BM model. Gruber & Köszegi (2001) reformulate the BGM94 work and confirm evidence for the adjacent complementarity hypothesis but not for time consistency of preferences1.
Forward-looking behaviour can be generated by time-inconsistent
preferences of the hyperbolic discounting type or by ‘cue driven’ behaviour. Therefore choice between models on the basis of such empirical tests seems difficult. Yet there are differences in the normative implications of the different models. The newer hyperbolic discounting and cue-driven models imply a case for policy based on internalities since perfect foresight then does not hold.
Indeed, the BM model assumes that users commence drug use with perfect foresight of the consequences of addiction. There is no ‘failure of will’. Agents choose what, for them, is their unique best option at any time prior to use of addictive goods and stick with that. Constant exponential discounting implies time consistent decisions where ‘weaknesses of will’ problems do not arise2. That an individual’s intertemporal choices can be modelled in such a way is a general hypothesis regarding consumer choice, introduced by Samuelson (1937), who was, however, cautious about the validity of this hypothesis and stressed its mathematical arbitrariness. The specific difficulty in an addiction context is that ‘impulsiveness’ is often part of the addiction experience. Addicts often indicate that their initial decision to use was a mistake that they would not repeat. This regret does not arise in a RA setting.
There are other concerns with the BM model:
(i) BM rely on high enough discount rates to provide an explanation for why people initiate addictive drug use given the well-recognised harms that are implied. BM thus sees individuals as inheriting a certain ‘degree of addiction’. For many this will be too 1
One setting where these criticisms were examined was with respect to pre-announced tax increases on cigarettes. Gruber & Köszegi (2001) shows such forecasts did reduce current consumption supporting the hypothesis of forward-looking behaviour. Also this finding was robust to specification tests. However the finding was also consistent with time inconsistent behaviour and hyperbolic discounting so that it did not necessarily support the BM formulation. 2 The time consistency implied by exponential discounting was noted by Samuelson (1937, p. 160).
7 low for it to be rational to continue to consume an addictive product indefinitely. For a few it is high enough to motivate continued and perhaps increasing use. Reliance on a high discount rates as a rationale for addiction can be relaxed using alternative addiction mechanisms such as the ‘Primrose Path’ model of Herrstein & Prelec (1992) and the Orphanides and Zervos (1995) model. (ii) The BM model supposes the degree of addiction decays exponentially so ceasing an addiction eventually leads to a cessation in demands for use. Therefore, if a user quits using, long-term cravings to use addictive products do not persist unless the stock of addiction depletes slowly or never depletes. But if it never depletes then an agent will not easily undergo withdrawal. Yet addicts do withdraw and, having done so, relapse into addiction is a common occurrence that is incompatible with a depleted stock of addiction. The more realistic picture is that agents can cease their use by withdrawing but still face persistent and intermittent long-term urges to reuse. (iii) Forward looking behaviour is difficult to apply to consumers who start using drugs in early teens - a common occurrence. It is known that the pre-frontal cortex region of the brain that is responsible for rational, forward looking behaviour is incompletely developed at ages up to the early twenties. It is hypothesised that consumption of addictive drugs at young ages inhibits the full development of the cortex area leading to poor judgements concerning drug usage even at older ages. For example there is strong evidence that early use of addictive drugs markedly increases the probability of excessive use at older ages: Koob et al. (2006, p. 8-10). (iv) For most people the rate of time preferences changes with age so the discount rate is typically lower for older consumers. Most young consumers fail to anticipate this. The RA model however assumes a fixed exponential rate of time preferences.
2.5 Policy implications. Given that the BM model posits rational self-interest with foresight there is, ignoring externalities, little reason to restrict supply of addictive substances by criminal sanctions. Such restrictions simply reduce social surplus. If restrictions are introduced there is an inevitable focus on price effects. The main finding of BM is that temporary changes in price have a smaller impact than sustained changes. We shall question and qualify this argument.
8
3. Solving the Becker-Murphy Model
The problem posed by Becker and Murphy (BM) is an optimal control problem with an integral constraint. It can be solved completely with strong enough assumptions.
Define the Hamiltonian function: H = e-σt u[y(t), c(t),s(t)] +ψ( c(t) − δs(t) − h[D(t)] ) + λ(A0/T+(w(s)–y–pcC–pdD)e-rt]
Using the Maximum Principle, necessary conditions for a maximum are:
∂H = 0 → u y = λe( σ− r)t ∂y ∂H = 0 → ψ = e − rt p c λ − e −σt u c ∂c λp d e − rt ∂H = 0 → h ′(D) = ∂D u c e−σt − p c λe − rt
(1) (2) (3)
.
ψ = [δ(p c u y − u c ) − u S − λw ′(s)e( σ− r)t ]e −σt
(4)
.
s = c − δs − h(D)
(5)
with transversality condition ψ(T)S(T)= 0
(6)
Setting θ =ertψ and assuming the discount rate equals the risk-free interest rate (r = σ) gives the more compact set of necessary conditions:
uy = λ
(7)
θ = pc λ − u c
(8)
h ′[D] =
pd λ u c − pcλ
(9)
θ '(t) = (r + δ)θ − u s − λw ′(s)
(10)
s '(t) = c − δs − h(D)
(11)
Given s(0) and the transversality condition (A6), e-rTθ(T)s(T) = 0 valid for T < ∞, these equations define optimal time paths. equations:
Rearranging and simplifying these
9
D '(t) = c '(t) = y '(t) =
−λ p d (u s − (r + δ ) θ + λ w '(s)) h ''(p c λ − u c ) 2 (c − h − δ s)(u cs u yy − u ys u yc ) + u yy (u s − ( δ + r) θ + λ w ') (u 2yc − u cc u yy ) (u ys u cc − u cs u yc )(c − h − δ s) + (u s − ( δ + r) θ + λ w ')u yc
s '(t) = c − h − δ s
(u 2yc − u cc u yy ) where
θ = p c u y -u c .
(13) (14) (15) (16)
The steady state of these equations defines an equilibrium in D, c, y, s given the multiplier λ which is determined from the budget constraint. In the special case where consumer preferences are quasi-linear in the non-addictive consumption good (so u(y,c,S) = u(c,S) + y), λ = 1 from (7) so (15) can be eliminated with the remaining three equations determining steady state c, S and D as:
u s − (δ + r)(p c − u c ) + w[s]' = 0
(17)
c − h[D] − δs =0
(18)
h ′[D] (u c − p c ) = p d .
(19)
With quasi-linearity the wealth constraint is eliminated because equipping the consumer with extra wealth only involves them spending more on the non-addictive good. This eliminates wealth effects on the demand for the addictive good that do play a role in the BM analysis though at the cost of additional complexity.
Adjustment paths to steady state equilibria following price changes will be considered once the complete dynamic optimisation task is analysed. This task can only be analysed qualitatively using phase planes analysis if the dimensionality of the problem is simplified, by suppressing any withdrawal dynamics.
In the special case where nothing is spent on rehabilitation (D = 0) equations (7) - (11) can be written more compactly as uy = λ = 1 and:
10 θ = pc − u c
(20)
θ'[t] = (r + δ)θ − w ′[s] − u s
(21)
s'[t] = c − δs
(22)
which, by eliminating θ, can be solved to yield the coupled equations (A22) and:
c '[t] =
u cs u (r + δ) w ′[s] (δs − c) + (u c − p c ) + s + u cc u cc u cc u cc
(23)
The properties of the level curve of (23) are difficult to specify without strong restrictions on u.
Following BM suppose that u can be locally described by a
quadratic function of c and s so: u(c,s) = αc.c + αss + αcc.c2/2 + αss.s2/2 + αcs.c.s
(24)
where uc = αc + αccc + αcs.s > 0, us = αs + αsss + αcs.c < 0, ucc = αcc < 0, uss = αss < 0, ucs = αcs >0. Further, suppose w can be described locally as: w[s] = wsss2/2 + wss +w0 where ws < 0 and wss < 0. Then (23) can be written: c '[t] = A + Es + (r + δ)c. where A =
(25)
(r + δ)(α c − p c ) + α s + w s α + (2δ + r)α cs + w ss and E = ss α cc α cc
The phase plane of (22) and (25) can be plotted. The locus s΄ = 0 has slope ∂c/∂s = δ > 0 and passes through the origin. The locus c΄ = 0 has slope ∂c/∂s = − (α ss + 2δα cs + rα cs + w ss ) α cc (r + δ) positive if and only if:
with sign( α ss + 2δα cs + rα cs + w ss ). This is
11 α cs > − (α ss + w ss ) /(r + 2δ)
(26)
and is non-positive otherwise. Note too that c΄ = 0 has a positive intercept with the c axis when s = 0 if A > 0 which is the case when:
(αc-pc)(r+δ) + αs + ws > 0
(27)
and has a negative intercept with the c-axis if A < 0 when the reverse strict inequality holds.
Moreover, comparing the slope of s΄ = 0 with that of c΄ = 0 we see that s΄ = 0 has a steeper slope than c΄ = 0 if and only if:
δα cc (r + δ) + α ss + w ss + (2δ + r)α cs < 0
(28)
This last condition holds unless αcs is strongly positive in which case consumption displays strong adjacent complementarity. Four situations arise:
(26) and (28) each hold
Case 1
c΄ = 0 positive-sloped, s΄ = 0 steeper with positive slope
stable case Case 2
(26) holds but (28) does c΄ = 0 positive-sloped and steeper
unstable case
not hold
Case 3
(26) does not hold but c΄ = 0 negative-sloped and s΄ = 0
stable
but
no (28) holds
than s΄ = 0.
steeper
adjacent complementarity Case 4
no steady state in the interior of the positive orthant R2++
12 Differentiating (22) and substituting c′[t] from (25) gives a second-order linear differential equation in s:
s" = c '− δs ' = A + Es + (r + δ)(s′ + δs) − δs′ so. or
s′′ − rs′ − Bs = A
(29)
where
B = E + δ(r + δ) =
αss + 2δα cs + rα cs + w ss + δ(r + δ)α cc α cc
and where B can potentially be positive or negative. The characteristic equation associated with this has solutions λ = (r ± r 2 + 4B) 2 with r2+4B > 0, given the assumed concavity of the utility function3. Hence the equation has two distinct and real eigenvalues λ1 < λ2. Note that the relevant transversality condition (6) for the case of quasi-linear utilities for large T can be written: e-rTθ(T)s(T) = e-rT(pc-uc)S(T) ≈ -e-rTc(T)S(t) ≈ -e-rTs(T)2 = 0.
(30)
Case 1: Here λ1 < 0 and λ2 = 0.5[r + r 2 + 4B] > r/2 > 0. As T becomes large the transversality condition (30) e-rTs(T)2 → e(2 λ2 − r )T so (30) cannot be satisfied. Thus the solution to (29) is of form s[t] = (s 0 − s*)eλ1t + s * where λ1 = 0.5(r − r 2 + 4B) with s* = - A/B, is the steady state optimal addiction stock and s0 the initial stock. Note with current restrictions there is no guarantee that s* will be positive
From (16):
3
u(c,s) is strongly concave in s if its Hessian is negative definite. Here r2+4B = {(r2+4δ(r+δ))αcc+4αss + 4(r+2δ) αcs +4wss}/ αcc= {(r+2δ)2αcc +4αss+4(r+2δ)αcs+4wss}/ αcc which is the sum of a quadratic form of type αccx2+αssy2 +2αcsxy where x=r+2δ, y = 2 which is negative if u is concave plus a term awss also negative. These two negative terms are divided by αcc also negative so r2 + 4B > 0.
13
c = s' + δs =λ1deλ1t + δs = λ1 ( s − s *) + δs = ( δ + λ1 ) s − λ1s * so c [t]= ( δ + λ1 ) s[t] − λ1s *
(31)
so the relation between c and s on an optimal time path is linear. Moreover, when c[t] = 0 we have s[t] =λ1s*/(δ+λ1) > 0 so the intercepts in the phase plane can be signed. Now note that (28) implies that B > 0. Thus if:
δα cc (r + δ) + α ss + w ss + (2δ + r)α cs < 0 then {δα cc (r + δ) + αss + w ss + (2δ + r)α cs }
α cc
>0
so B > 0.
Finally note (26) implies that the approach path (31) has a positive slope. Thus if:
α cs > − (α ss + w ss ) /(r + 2δ) then (r + 2δ)α cs + (α ss + w ss ) + δ(r + δ)α cc > δ(r + δ)α cc so
4{(r + 2δ)α cs + α ss + w ss }
α cc
+ 4δ(r + δ) > 4δ(r + δ)
or 4B < δ 2 + rδ ⇒ r 2 + 4B < 4δ 2 + r 2 + 4rδ
On taking positive square roots and rearranging:
r − r 2 + 4B + δ > 0. 2
So λ1+δ > 0 and the locus (31) does indeed have a positive slope.
14 The phase plane describing the c, s dynamics are as in Figure 1 below for the interesting case where the terminal optimal stock of addiction is positive. c s' = 0
c' = 0
c*
s s* Figure 1: Case 1 dynamics
This corresponds to what BM term the ‘stable case’. Both s′ = 0 and c′ = 0 have positive slopes with the slope of s′ = 0 being larger. The unique steady state in this case is stable, and the path of convergence is linear and positively sloped. The consumers with low initial addiction levels will increase consumption each period and move to the steady state with a higher stock s*. Consumers with a large addiction level will consume less each period and move to a steady state with a lower stock of addiction s*.
Note that this case requires two conditions (26) and (28) to be satisfied. (26) corresponds to the condition (16) from BM – the adjacent complementarity condition that guarantees the positive slope of the convergence path. The other condition corresponds to B > 0 in BM and guarantees convergence to a stable steady state. The restrictions on the cross partial αcs resulting from (26) and (28) work in opposite directions. This parameter must be large enough (adjacent complementarity must be large enough) so that on the optimal path consumption increases with the stock of addiction, but small enough so the optimal path converges to a stable steady state.
15 The long-run comparative statics are illustrated in Figure 2. A sustained increase in the price of addictive consumption reduces steady state addictive consumption and the degree of addiction. The dynamics of a permanent (sustained) increase in price that is unanticipated up to the time it occurs are graphed assuming that a positive long-run optimal usage path remains optimal. Given a large enough price increase the optimal steady state addiction stock could be negative. Then it would be optimal to immediately respond to the shock by permanently abstaining from all future consumption (to go ‘cold turkey’) or to switch to a usage path that involves diminishing use for a phase and then permanent cessation of use.
In Figure 2 an increase in price lowers the c΄ = 0 curve from c΄old = 0 to c΄new = 0. The slope of this curve will not change with a price change4 if wages w are quadratic in the degree of addiction. Note that the slope and position of the s΄ = 0 line remains fixed with a price change. The approach path to the addiction equilibrium undergoes a parallel downward shift since only the constant term in this linear equation depends (negatively) on price. Three possible dynamic patterns of adjustment occur with respect to price depending on the size of the addictive stock at the time of the permanent price increase (say s1) relative to the new and old equilibrium addiction stocks (s*new, s*old). •
If s1 < s*new then, prior to the unanticipated shock, consumption and addiction evolve along the path I, consumption jumps down by the length of A and then, after the shock, the adjustment path is B. The short run reaction of the consumer is to reduce consumption of the addictive good substantially, but to increase it slightly afterwards.
•
Similarly, if s*new < s1 < s*old , then the adjustment path jumps from I΄ by the distance of A΄ to B. This case most of the decrease in consumption clearly can occur initially at the time of the jump with subsequent decrements being less important.
•
Finally, if s1 > s*old, the adjustment path switches from I˝ by distance equal to the length of A˝ to B˝.
4
The equation of (A25) =0 is A+Es+(r+δ)c = 0 so dc/ds = -E/(r+δ) < 0. This depends on the price of addictive consumption only through the term which depends on price through its effect on w˝(s). These wage effects vanish if w is quadratic in w.
16
s' = 0
c
c'old = 0
I''
I'
I
A''
c' new = 0
A'
A B B'
B''
s s*new
s*old
Figure 2: Comparative statics of a price increase
This figure allows us to discuss the relative size and importance of short-run and longrun effects. When the price change impacts on a consumer in steady state equilibrium the long run effect of the price change is larger than the short-run effect, as correctly claimed by BM. However if a consumer is affected by the price shock while approaching a steady state this need not be so. Then an increase in price will lead to an immediate large drop in the current consumption that will be either offset or increased in the long-run, depending on whether the consumer was on the path with increasing or decreasing consumption. The relative size of short-run and long-run effects depends on the slope of the convergence path: if the path is almost horizontal, so addictive consumption does not change much with addiction, the short-run effect is almost equal to the long-run effect. Then a rational consumer will adjust his consumption immediately in response to the permanent price change.
The higher the degree of adjacent complementarity and the addiction depreciation rate, the steeper is the convergence path and the larger are the long-run effects of a price change compared to the short-run effects.
17
Case 2. If (28) does not hold then B < 0. The eigenvalues of the characteristic equation associated with (29) are 0 < λ1 = 0.5[r − r 2 + 4B] < r/2 and λ2 = 0.5[r + r 2 + 4B] > r/2. The transversality condition (30) requires λ < r/2 so λ2 is ruled out. But the remaining root λ1 > 0, while consistent with transversality, implies an unstable adjustment path s = (s0 – s*)e
λt 1
+ s*. If (26) holds the adjustment path c =
(λ1+δ)s - λ1s* has positive slope. It is graphed as the unstable path in Figure 3.
c
c' = 0 s' = 0
s Figure 3: Case 2 dynamics
Note that the conditions for an unstable adjustment path to arise are restrictive. For B < 0, (28) cannot hold so α cs > − (δα cc (r + δ) + α ss + w ss ) /(2δ + r) . For concavity of u we require α cs ≤ α cc α ss . Combining these establishes the need for strict upper and lower bounds on αcs which determine the extent of adjacent complementarity in addictive consumption.
αcs must be high enough to generate high intertemporal
complementarities in consumption but not so high as to violate the concavity condition. Specifically for an unstable equilibrium:
α cc α ss ≥ α cs > − (δα cc (r + δ) + α ss + w ss ) /(2δ + r) .
(32)
18 Thus with r = δ = 0.2, αcc = -1, αss= -0.08, wss = 0, (A32) requires 0.28 ≥ αcs ≥ 0.27 a narrow band.
The unstable situation can be characterised thus. Denote the unstable equilibrium state (c*, s*). If the initial stock of addiction s*(0) > s*, then optimal initial consumption c*(0) > c*, and the extent to which users consume grows without bound as does their level of addiction. If s*(0) < s*, then c*(0) < c* and users begin using but at a rate which cannot sustain their level of addiction. Their usage rate falls to zero whereupon the level of addiction falls monotonically towards zero as well.
BM place considerable weight on the existence of the unstable steady states. To BM they are ‘crucial to the understanding of rational addictive behaviour’. For example, the more addictive a good is the greater the likelihood of instability. The unstable steady state is especially important for explaining how small permanent change in price could lead to a significant long run effect on consumption.
Case 3. Finally, consider the situation where the level curve of c΄= 0 is negatively sloped and s΄ = 0 is steeper. Adjacent complementarity condition is not satisfied in this case. See Figure 4.
c s' = 0 c' = 0
s Figure 4: Case 3 dynamics
19
Again there is a stable branch of the saddle point of these dynamics that leads to long run equilibrium. The indicated pattern of stable adjustments here however looks much less plausible since, along any such path, consumption either falls as the degree of addiction increases or consumption increases as the level of addiction declines.
4. The general nonlinear case
BM claim that the results provided for a quadratic utility suggest three features of what could happen with a more general nonlinear utility: (i) the situation where there are two steady states (one stable, one unstable) and where the unstable steady state corresponds to a lower addiction level than the stable one; (ii) ‘cold-turkey’ behaviour, and (iii) ‘binging’ behaviour. These situations are now discussed in turn.
4.1 Multiple steady states. BM emphasise in their analysis a situation where there are two steady states (one stable, one unstable) and where the unstable steady state corresponds to a lower addiction level than the stable one. An unstable low equilibrium steady state might arise in a way comparable to that observed in Figure 3 with agents either driving their consumption to zero or else increasing their consumption until they converge to a stable steady state as described in Figure 1. BM use this to rationalise a potentially bimodal distribution of users some of whom consume none of an addictive good while others consume much larger amounts. They also suggest that with this configuration of equilibria that long-run effects of events such as price changes might be significantly larger than short-run effects.
As was discussed above, the existence of the unstable steady state is rather unlikely; so this will necessarily be rather a special case. But even if there are both stable and unstable equilibria the short-run and long-run effects of price on consumption depend significantly on which of the equilibria (stable or unstable) corresponds to a lower level of addiction as well as the distribution of consumers across different levels of addiction.
20 Consider first the BM situation where the unstable steady state corresponds to a lower level of addiction than the stable state and consider the effects of a price increase. This situation is described in Figure 5. c s'=0 c'old=0 c'new=0
p1 p2
s s 1L
s 2L
s 2H
s 1H
Figure 5: Strong effects of a small price increase
Assume that most consumers have a degree of addiction in the neighbourhood of the unstable steady state s1L in Figure 5 and that only a relatively small number of users are heavily addicted in the neighbourhood of the stable steady state s1H. What is the effect of a sustained change in price of the addictive good on consumption in this case? The price shock shifts the c΄ = 0 curve downwards but leaves unchanged the s'=0 curve. The optimal addiction time path switches from p1 to p2. Then a relatively small price increase will lead heavy addicted people (whose addiction levels are initially near s1H to cut their consumption a little until they reach the new equilibrium addiction level s2H. But those consumers with consumption just above the unstable steady state (those whose addition lies between s1L and s2L) who were initially destined to become heavy users by moving towards the high addiction steady state s1H will now decrease consumption and eventually quit consuming altogether. In this case therefore a small permanent increase in the price of addictive good may lead to dramatic decreases in consumption and equilibrium levels of addiction.
21
If the steady states are reversed, so that the stable steady state corresponds to a low rather than the high level of addiction then, from Figure 6, the price increase will have almost no effect on the people with low addiction level, but lead some people with high addiction level (those with addiction between s1H and s2H) to move toward much lower addiction levels. It is reasonable to suppose most potential drug addicts have rather low levels of addiction initially so they are located near s1L. Only a few are located near the unstable steady state s1L because most long-time addicts will move to much higher addiction level than s1L. Now a small sustained increase in price will have almost no significant effect on aggregate consumption of the addictive good. Clearly the analysis of relative addiction levels for stable and unstable steady state consumption is crucial for understanding addictive behaviour. There is no presumption here favouring the BM view. Indeed, for the utility function
u ( y, c, s ) = y + ln(c + 1) + α cs − β s 2
for all values of parameters α and β, where there are exactly two steady states, the steady state with a lower stock of addiction is always the stable one – the result opposite to that envisaged in the BM scenario. (See Gavrila et al. (2005) for proof).
22 c p2 p1 s'=0
c'old=0
c'new=0
s s 2L
s 1L
s 1H
s 2H
Figure 6: Weak effects of a small price increase
4.2 Cold turkey behaviour. ‘Cold turkey’ behaviour means abrupt cessation of use of an addictive good when an agent is already highly addicted. Drug addicts, for example, often find it difficult to decrease their use gradually over time and instead prefer to go ‘cold turkey’ to end drug use. BM claim that small changes in external events, such as price increases, can motivate this type of behaviour in a RA setting so ‘Weak wills and limited self-control are not needed to understand why addictions ……can end only when the consumption stops abruptly’ (ibid p.693 ).
However BM do not explain cold turkey using their variant of the RA model with one stable and one unstable steady state as described in Figure 5. Instead they introduce a non-concavity into the utility function. Specifically the cross partial derivative of utility with respect to consumption and the addiction stock is required to be so large (called strong adjacent complementarity) that it violates concavity even though second derivatives of utility with respect to consumption and the stock are negative.
As a result, roots of the characteristic equation describing the addiction dynamics are complex and ‘jump’ behaviour can become optimal. For levels of addiction below
23 some critical stock level BM claim that minimal (zero) consumption is desirable while for addiction levels above the same stock, maximum feasible consumption is optimal.
This seems to be an unsatisfactory explanation of the ‘cold-turkey’ phenomenon. Consider the case where the cross partial derivative of the utility function with respect to consumption and the addiction stock become so large that it violates concavity. Then the unstable steady state (node) becomes the focus. The optimal adjustment path will have discontinuity at some critical level of addiction between the lower stable steady state and the focus steady state. Consumers with an initial addiction stock lower than the critical level will converge to the steady state, as in concave case while those consumers with addiction stocks above the critical level will increase consumption and stock indefinitely as in the unstable concave case. This is illustrated in Figure 7. A specific example is provided in Gavrila et al. (2005).
c
s'=0 unstable steady state (focus)
stable steady state (saddle) c'=0
s s 1L
sc
s 1H
Figure 7
Therefore, even if a discontinuity in the path exists, it is not observed in actual consumption behaviour so there is no cold turkey type withdrawal. Consumers to the
24 left of sc converge to the stable steady state while consumers to the right of sc increase their consumption indefinitely.
The small change in price may affect the critical level of stock only slightly (or not at all as in the quasi-linear utility case). As a result, the jump in consumption, though not impossible, is very unlikely. Note also that if the critical level of addiction decreases with the price change, it might be optimal for the consumer to ‘jump up’ instead of quitting cold turkey. Then the consumer will increase consumption in the short-run significantly.
4.3 Binging behaviour. BM account for ‘binging’ behaviour of the type that occurs with the bulimic consumption of food by adding to their underlying model an extra stock variable (e.g. ‘weight’) complementary with the degree of addiction variable (‘hunger’).
Hunger is complementary with eating while weight is substitutable.
Weight has a lower depreciation rate than hunger since it takes time to lose weight. Then a thin person who becomes addicted to food will acquire more hunger than weight as hunger depreciates faster. Ultimately eating will fall as weight continues to increase. Lower food consumption reduces hunger relative to weight and the reduced hunger keeps eating down even as weight decreases. Eventually weight gets to a low level when eating resumes again and the cycle begins again. This then explains cycles in consumption.
But binge behaviour can be readily explained, even when a single stock drives addictive consumption, by accounting for the possibility of reducing addictive behaviour through what BM term ‘endogenous depreciation’ or ‘investment in withdrawal’.
Thus if the cost of withdrawal is low a consumer will find it optimal to get addicted knowing that it is easy to reverse this decision by investing in withdrawal. As a result, binge behaviour emerges. The consumer can commence using drugs, knowing that the addiction can be terminated (at some cost) later when the addiction level is high. If it is feasible for the consumer to restore the original degree of addiction they will find that they are in exactly the same position as initially, creating incentives to repeat the experience. There is, in this case, an optimal cycle of binge behaviour. If
25 the cost of combating the addiction is high enough then binge behaviour will no longer be optimal. Indeed, as a result, some consumers will chose not to consume at all after accounting for the costs of withdrawal.
Figure 8 provides a discrete time variant of the BM model. It graphs instantaneous consumption utility over time when an addictive good is optimally consumed less the instantaneous optimised utility that is derived from, consuming non-addictive goods alone. Addictive consumption begins at time t1 and, up to t1, a user not consuming the addictive good is experiencing low utility relative to the utility that is subsequently realised from addictive consumption. With addictive consumption, starting at t1 through to t2, net additional utility from addictive consumption remains positive but after t2 it becomes negative due to the direct negative ‘tolerance’ effects of addiction on utility. Consumption over (t1, t2] is rational only because the consumer’s discount rate is high enough to nullify the effects of the future disutilities of sustained addictive consumption.
Addictive consumption utility less abstinence utility
t2
t3
t4
0 t1
Disutility
Figure 8: Utility Time Path from Addiction
26 Discount rates play a role with respect to the decision to cease addictive consumption. Suppose at time t3 in Figure 8 the user has a discrete option to cease use of the addictive good by ‘withdrawing’ and experiencing a sharp negative disutility. As
Figure 8 is drawn withdrawal is optimal with the agent subsequently returning to close to his pre-addictive consumption utility level. This propensity to cease use presupposes that the agent’s discount rate is not excessively high since, otherwise, the agent will not be prepared to experience the high short-term costs associated with the withdrawal experience to gain access to lower costs further into the future.
Note too that if an agent’s discount rate is high enough, addiction is initiated more readily and withdrawal from an addiction is less likely since it will be less beneficial. This means that reducing the costs of withdrawing from a drug increases the incentives to initiate use and increases the scale of use once it is initiated. Note, finally, that if withdrawal involves reducing the stock of addiction back to its initial level s(0), addiction will be reinitiated in a stationary environment and that in fact recurrent cycles of addiction will make sense even though there is a single stock of addiction. For example ‘binge eating’ among bulimics may be less of an issue of selfcontrol than the perception that the adverse effects of gorging on food can be readily undone by purging if the latter is judged to involve low enough costs5.
This is a particular problem for harm minimization strategies that reduce the cost of withdrawal. As a result more people will use drugs both because it is safer and less costly to do it now and because they can terminate the addiction at relatively low cost in the future if they want to. Policies for dealing with addiction therefore have to resolve trade-offs between helping the people currently addicted and making drug use more attractive.
A general difficulty with the BM’s results is that they are not derived from the model analysed. It is not shown that the two types of equilibria analysed in the quadratic utility model are local equilibria of a more complete non-linear model or indeed that the latter class of models has two rather than three or more equilibria. The rationale for ‘cold-turkey’ withdrawals and for ‘binging’ behaviour therefore cannot be linked 5
Thus the claim by BM that ‘two capital stocks are needed to get cycles’ seems incorrect if account is taken of the possibilities for withdrawal.
27 to a nonlinear model with non-concavities for the same reason.
To add to the
confusion if it is to be claimed that ‘cold turkey’ withdrawal can be induced by an upward price shock, as we have already shown in our analysis of price increases, a large enough increase in price can make the optimal addiction stock negative. In this case ‘cold turkey’ behaviour occurs without introducing non-concavities into the utility function.
5. Empirical work
A problem with empirical confirmation of the RA model is that much empirical work is based on the quadratic version of the model, which cannot generate the two steady states used by BM to explain effects of price on consumption. Binge behaviour and cold turkey behaviour cannot be explained either. If empirical evidence supports the quadratic case with one (stable or unstable) steady state or the basic case with one stable and one unstable SS, then this would not be consistent with binge or cold turkey behaviour that requires an extra stock variable (for binge behaviour) and nonconcavity (for cold-turkey behaviour).
Moreover many of the empirical conclusions of the RA model are consistent with other models and approaches. Although there is empirical evidence that consumers are forward-looking, many models assume forward-looking behaviour so that demand for addictive good depends on both past and future prices as suggested by RA. Therefore, it is difficult to distinguish between RA model and other models, such as those based on hyperbolic discounting, from empirical evidence.
6. Conclusions
In this paper we revisited the RA model of BM. Using a formal optimal control approach we derived conditions for stability or instability of optimal steady states in terms of utility function parameters. We showed that the existence of the unstable steady state is unlikely because it imposes severe restrictions on model parameters.
Moreover, even if there are exactly two steady state equilibria as BM suggest their relative positions may differ from those hypothesised in BM. This can significantly
28 alter the analysis of price changes on consumption. In the BM case a small increase in price might lead to very significant long run effects whereas in the opposite case effects may be negligible. Thus, the analysis of the existence and relative position of steady states is crucial for understanding the addictive behaviour and for the policy recommendations.
We show that the cold turkey behaviour is not satisfactory explained by BM and that binging behaviour is easier to explain by taking into account the cost of quitting instead of introducing two different stocks of addiction.
Finally empirical evidence that suggests forward-looking behaviour is consistent with a number of alternative models of addiction.
In conclusion, further theoretical and empirical research is required to determine whether the rational addiction or other competing models explain addictive behaviour better and should therefore be used as the basis for policy recommendations.
29
References
G. S. Becker & K. Murphy, ‘A Theory of Rational Addiction’, Journal of Political
Economy, 96, 1988, 675-700.
G.S. Becker, M. Grossman & K. Murphy, ‘An Empirical Analysis of Cigarette Addiction’, American Economic Review, 84, 3, 1994, 396-418.
G.S. Becker, M. Grossman & K. Murphy, ‘Rational Addiction and the Effect of Price on Consumption’, American Economic Review, 81, 2, 1991, 237-241.
C. Gavrila, G. Feichtinger, G. Tragler, R.F. Hartl, P.M. Kort, ‘History-dependence in a Rational Addiction Model’, Mathematical Social Sciences, 49, 2005, 273-293.
J. Gruber & B. Köszegi, ‘Is Addiction ‘Rational’? Theory and Evidence, Quarterly
Journal of Economics, November, 2001, 1261-1303.
G.F. Koob & M. Le Moal, Neurobiology of Addiction, Academic Press, Amsterdam, 2006.
A. Orphanides & D. Zervos, ‘Myopia and Addictive Behaviour’, Economic Journal, 108, 1998, 75-91.
A. Orphanides & D. Zervos, ‘Rational Addiction with Learning and Regret’, Journal
of Political Economy, 103, 1995, 739-758.
