A Theory of Rational Addiction

The paper’s definition is that “a person is potentially addicted to c if an increase in his current consumption of c increases his future consumption of c,” and this occurs if and only if his behavior displays adjacent complementarity (Sec. IV, p. 681). Formally, utility at each moment depends on consumption of good c, a non-addictive good y, and a stock of “consumption capital” S that accumulates past consumption of c net of depreciation (Sec. II, eqs. 1-2). Adjacent complementarity holds when the rise in the marginal utility of c from a higher S exceeds the rise in the “full price” of c caused by the same higher S (eq. 16); the paper shows this condition is necessary and sufficient both for potential addiction and, separately, for present and future consumption of the good to be complements in the ordinary (Slutsky) sense (Sec. VI, p. 689) – a point the authors present as a correction to Ryder and Heal’s (1973) claim that adjacent complementarity was analytically distinct from Slutsky complementarity.

Unstable steady states are, in the authors’ words, “a major tool in our analysis of addictive behavior”: consumption rises over time when it starts above an unstable steady-state level, and falls over time – potentially to abstention – when it starts below that level (Sec. VIII, p. 694). The model’s first-order conditions yield a steady state in c and S whose stability is governed by a parameter B (eq. 11); the steady state is stable if B > 0, and the paper shows that B turns negative – making the steady state unstable – once the degree of adjacent complementarity (determined by the rate of time preference, the depreciation rate of consumption capital, and the cross-partial of utility in c and S) is sufficiently strong (Sec. IV, pp. 682-683). The authors stress that instability is not an incidental technical feature to be assumed away: it is “crucial to the understanding of rational addictive behavior,” because even small initial deviations from an unstable steady state generate large, self-reinforcing cumulative changes in consumption, explaining why addiction can appear to take hold rapidly.

Quadratic utility and earnings functions alone cannot generate multiple steady states, but the paper shows that adding a negative cubic term in S to the utility function (so that the degree of adjacent complementarity declines as the stock S grows) produces two interior steady states, one stable and one unstable (Sec. IV, pp. 683-684). With two steady states, the authors argue, “relatively few persons consistently consume small quantities of addictive goods,” because consumption diverges away from the unstable level toward either zero or the sizable stable level – so “goods that are highly addictive to most people tend to have a bimodal distribution of consumption, with one mode located near abstention,” and the paper offers cigarette and cocaine consumption as examples of this bimodality, contrasting them with the more continuous distribution of alcohol consumption, which the authors attribute to alcohol “not [being] addictive for many people” (p. 684).

Because present and future consumption of an addictive good are complements, a (compensated) increase in its future price lowers current consumption, so a change confined to the current period has a smaller effect on current consumption than a permanent change that also raises future prices (Sec. VI, pp. 688-689). With a quadratic utility function, the paper derives that the long-run change in steady-state consumption from a permanent price change (eq. 18) grows in magnitude as the degree of addiction increases (since greater addiction lowers the stability parameter B in the denominator), so that “the long-run demand for addictive goods tends to be more elastic than the demand for nonaddictive goods” even though “permanent changes in prices of addictive goods may have a modest short-run effect” (Sec. VIII, p. 695). The paper cites reviewed cigarette-demand elasticity estimates clustering between 0.4 and 0.5 (Mullahy 1985) and long-run estimates around 0.6 from models that implement the authors’ own framework (Becker, Grossman, and Murphy 1987), noting these are “not small compared to elasticities estimated for other goods,” and cites alcohol elasticities reviewed by Cook and Tauchen (1982) as appearing higher, especially for liquor (Sec. V, p. 686).

The authors treat the diffusion of information on smoking and health beginning in the late 1950s as “an excellent experiment on whether persons addicted to smoking consider delayed harmful consequences,” and cite Ippolito, Murphy, and Sant’s (1979) estimate that per-capita cigarette consumption and tar/nicotine intake fell 34 percent and 45 percent respectively in the 11 years after the 1964 Surgeon General’s report – evidence the authors say “blatantly contradicts the view that the majority of smokers were myopic” (Sec. V, pp. 687-688). The teenage evidence is presented as a sharper test: because teenagers are often characterized as among the most impatient (citing questionnaire evidence from Davids and Falkoff 1975), a myopic model predicts their smoking should be little affected by health consequences delayed 20 or more years, yet smoking rates among males aged 21-24 “declined by over one-third from 1964 to 1975” (citing Harris 1980) – a decline the authors read as showing that even relatively present-oriented smokers respond to anticipated future costs.

The paper derives that whether an addictive good is a “superior” or “inferior” good as wealth rises (eq. 19) depends on how an increase in earnings changes the marginal disutility that higher consumption capital imposes on earnings; for harmful goods this term is likely positive because “the negative effect on earnings of increased consumption is likely to be greater when earnings are greater,” giving the example that heavy drinking reduces an airline pilot’s or doctor’s productivity more than a janitor’s (Sec. V, p. 688). The authors argue that the spread of information about smoking’s health hazards should have reduced, and could even have reversed the sign of, the income elasticity of smoking demand, and note this elasticity “did decline after the 1960s to a negligible level” (citing Schneider, Klein, and Murphy 1981); they further suggest that because women earn less than men on average, this earnings-linked channel may help explain why smoking by women grew relative to smoking by men over the subsequent 25 years (p. 688) – offered as a suggestive implication rather than a tested result.

The paper models stressful events as raising the marginal utility of the addictive good (equivalent, in its structure, to a temporary favorable shift in relative prices), and shows that a temporary rise in demand can move a person past an unstable steady state so that, when the stress ends, the person “accumulates sufficient capital while under stress to remain hooked afterward” even though, absent the stress episode, “he would have eventually abstained” (Sec. VI, pp. 690-691). The authors apply this directly to the empirical pattern that “although most Vietnam veterans did end their addiction to drugs after returning to the United States, many did not, and others shifted from dependence on drugs to dependence on alcohol” (citing Robins et al. 1980), treating this as consistent with – not as a counterexample to – a rational, forward-looking model, since the underlying mechanism (an unstable steady state crossed during a temporary shock) is the same one driving the paper’s other results.

The authors address the criticism (attributed to Winston 1980 among others) that a rational-choice model implausibly implies addicts are “happy,” responding that their model “recognizes that people often become addicted precisely because they are unhappy” when addiction originates from utility-reducing events like death or divorce – while insisting such people “would be even more unhappy if they were prevented from consuming the addictive goods” (Sec. VI, p. 691). The authors distinguish this from approaches (Yaari 1977; Elster 1979; Winston 1980; Schelling 1984) in which “different preferences or personalities fight for control over behavior,” such that a person’s “nonaddictive personality” tries to pre-commit against an “addictive personality” (e.g., by joining Alcoholics Anonymous): in their model, by contrast, “present and future consumption of addictive goods are complements,” so increased addiction when future consumption is expected to rise reflects “both present and future behavior [as] part of a consistent maximizing plan” rather than a struggle between competing selves (p. 692).

The paper shows that once the degree of adjacent complementarity is strong enough to make the joint utility function in c and S non-concave (though each remains concave separately), the optimal relationship between current consumption and the capital stock becomes discontinuous at a critical stock level S̃, with consumption high just above the threshold and low just below it (Sec. VII and App. B, pp. 692-693, 696-697). The economic logic is that when S is just above S̃, the good is addictive enough that optimal future consumption – and hence, through strong complementarity, optimal current consumption – is high, whereas just below S̃ future consumption is low because the addiction is expected to end quickly, so current consumption should also be low; the authors conclude that “with a sufficiently large discontinuity, an addict who quits would use cold turkey; that is, he would immediately stop consuming once he decides to stop,” and that this is true even though “the short-run loss in utility from stopping consumption gets bigger as an addiction gets stronger” – rational persons accept the large short-run cost because it is traded for an even larger long-run gain, with “weak wills and limited self-control” not required to explain the pattern (Sec. VII, p. 693).

A single consumption-capital stock cannot generate cyclical “binge” behavior; the paper extends the model to two stocks – for overeating, “weight” and “eating capital” – with different depreciation rates and different complementarity/substitutability with current consumption, which is sufficient to generate cycles (Sec. VII, p. 693-694, App. C). In the overeating example, eating capital must be complementary with eating and have the higher depreciation rate, while weight (the slower-moving, substitutable stock) eventually drags consumption back down once it accumulates enough; the resulting cycles “can be either damped or explosive (or constant) depending on whether the steady state is stable or unstable.” The authors emphasize that, in their framework, binges “do not reflect inconsistent behavior that results from the struggle among different personalities for control” but are instead “the outcome of consistent maximization over time that recognizes the effects of increased current eating on both future weight and the desire to eat more in the future” (p. 694).

The authors describe their analysis as building on “the model of rational addiction introduced by Stigler and Becker (1977) and developed much further by Iannaccone (1984, 1986),” and as relating to the habit-persistence literature of Pollak (1970, 1976), Ryder and Heal (1973), Boyer (1978, 1983), and Spinnewyn (1981) (Sec. I, p. 676). They credit Boyer (1983) and Iannaccone (1986) with first linking addiction to adjacent complementarity, but distinguish their own contribution: “We appear to be the first to stress the importance for addictions of unstable steady-state consumption levels, to derive explicit long- and short-run demand functions for addictive goods, to show why addictions lead to abrupt withdrawals and binges, and to relate even temporary stressful events to permanent addictions” (p. 676). They also note that Boyer’s (1983) discrete-time special case, in which the stock equals lagged consumption exactly, collapses the distinction between adjacent complementarity and the direct cross-partial effect of S on marginal utility of c that the continuous-time model keeps analytically separate (Sec. IV, p. 682).

The authors are explicit that they “do not claim that all the idiosyncratic behavior associated with particular kinds of addictions are consistent with rationality,” presenting the paper’s claim instead as that a rational-choice model “does explain well-known features of addictions and appears to have a richer set of additional implications about addictive behavior than other approaches” (Sec. VIII, p. 695). They also flag an internal tension in their own definition of rationality: because a sufficiently high rate of time preference formally satisfies their rationality condition even though it implies near-total disregard of the future, the authors ask directly, “should someone who entirely or largely neglects future consequences of his actions be called rational?” – noting that “some economists and philosophers even suggest that rationality excludes all time preference” – and offer, as a partial response, reinterpreting high time preference late in life as approximately rational myopia driven by a short remaining horizon rather than pure impatience (Sec. IV, pp. 683-684).