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Non-adherence to a low-fat diet: an economic perspective
Journal of Economic Behavior & Organization Vol. 48 (2002) 93–104
Non-adherence to a low-fat diet: an economic perspective Gideon Yaniv a,b,∗ a
National Insurance Institute, 13 Weitzman Avenue, Jerusalem 95437, Israel b College of Management, Tel Aviv, Israel Received 15 November 1999; accepted 16 October 2000
Abstract Despite bearing increased risk of suffering a heart attack, failure to adhere to a low-fat dietary regimen is wide-spread. The paper applies an expected–utility–maximization model to inquire into the individual’s decision to deviate from a prescribed low-fat diet, offering an economic explanation for excess high-fat consumption. The implications of self-protection against the risk of dying from an attack are further examined. The analysis reveals that self-protection and dietary adherence can be complements, suggesting that public health policy may help reduce both the risk of an attack and the risk of dying from an attack through providing incentives for self-protection. © 2002 Elsevier Science B.V. All rights reserved. JEL classification: I12; I18; D81 PsycINFO classification: 3360; 3361 Keywords: Low-fat diet; Non-adherence; Heart attack; Hazard rate; Self-protection
1. Introduction Coronary heart disease (CHD), caused by the narrowing of the coronary arteries that supply blood to the heart, is the number one killer in the US, accounting for 40% of all deaths. It is considered, in part, to be a disease of lifestyle, associated with a number of poor health habits, such as a high-fat, cholesterol-rich diet, smoking and a low activity level. A primary clinical manifestation of CHD is myocardial infarction (MI), otherwise known
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as a heart attack, which is the death of the heart muscle due to the loss of blood supply. 1 Approximately, 1 million Americans suffer a heart attack annually, 400,000 die as a result. Consequently, modification of CHD risk-related behavior to reduce the incidence of CHD morbidity and mortality obtain high priority in public health intervention efforts. 2 However, getting people to develop good health habits and give up the risky behaviors that damage their health is not an easy task. In particular, despite convincing evidence that adherence to low-fat diet can significantly lower blood cholesterol levels, 3 failure to adhere to dietary restrictions is wide-spread. Why is it then that high-risk individuals might not adhere to prescribed dietary regimens? Health psychology suggests a number of explanations: 4 preferences for meat and fat dairy products are well established and hard to alter; dietary recommendations might necessitate drastic changes in meal planning, cooking methods and eating habits; they may be too restrictive, monotonous, and associated with a lower quality of life. In addition, a low sense of self-efficacy (i.e. disbelief in one’s ability to change one’s diet), or a strong sense of invulnerability or fatalism may attenuate adherence. The present paper addresses this question from an economic standpoint, proposing a rational decision-making model to explain individuals’ deviation from a prescribed low-fat diet. Most importantly, while psychological explanations focus on habit, reduced life quality, or irrational beliefs as deterrents to adherence, the economic model takes explicit account of the risk involved, confronting future costs with present benefits of non-adherence. The analysis reveals that the risk of undergoing a heart attack, which is assumed to increase with the level of excess fat consumption, is not just a deterrent to non-adherence. It also plays the role of a discount factor of future costs, reducing their value more the greater the deviation from the recommended diet. This drives the individual to behave less respectfully towards the future, offering an economic motivation for a seemingly habitual or irrational behavior. Another interesting result is that an increase in the level of external risk factors (e.g. high blood pressure, diabetes, stress, family history), might increase high-fat consumption, conforming with the fatalistic notion that if one believes that his time is short, he will seek increased enjoyment from the time still left. The present paper belongs to as yet a small literature which applies the rational economic approach to analyzing traditional health/clinical psychology phenomena, such as compulsive consumption (Winston, 1980), addictive behavior (Becker and Murphy, 1988), suicide threat (Hamermesh and Soss, 1974), or fear of public places (Yaniv, 1998). The paper begins (Section 2) with modeling the individual’s decision to consume high-fat products in 1 A heart attack most often occurs due to the accumulation of cholesterol (a fatty chemical found mainly in animal products) on the inner walls of an artery to the heart. Over time, the build-up of cholesterol deposits may cause substantial thickening of the artery walls and significant narrowing of the artery. Occasionally, the cholesterol plaque becomes sticky, triggering a blood clot to form on its surface that completely blocks the artery (American Heart Association, 1993). 2 In 1985, the National Institute of Health (NIH) launched a national campaign known as the National Cholesterol Education Program (NCEP), with the purpose of reducing illness and deaths from CHD through encouraging the general public to go on a low-fat diet. Recent surveys show that from 1985 to 1995, the percentage of Americans who had ever had their blood cholesterol checked rose from 35 to 75 and that blood cholesterol levels among US adults dropped, on average, from 213 to 203 mg/dl. 3 See Carmody et al. (1987) for a review of the literature. 4 See Taylor (1995, Chapter 5) for a detailed discussion of the psychological explanations and related literature.
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excess of a prescribed low-fat diet, risking the occurrence of a heart attack at some uncertain time in the future, the distribution of which is affected by excess fat consumption. It then proceeds (Section 3) with examining the implications of self-protection against the risk of dying from an attack, through subscribing to emergency services that reduce delay in diagnosing an attack or in obtaining emergency treatment. The analysis reveals, counter-intuitively, that self-protection and dietary adherence can be complements, suggesting that public health policy may help reduce both the risk of an attack and the risk of dying from an attack through providing incentives for subscribing to delay-reducing services. The paper concludes (Section 4) with a summary of the main results and some related remarks.
2. The model Consider an individual, who, during a given period of time, spends his disposable income, Y, on the consumption of high-fat, cholesterol-rich products, c, and low-fat, cholesterol-free products, h. Suppose that a blood test taken at the beginning of the period reveals abovenormal values of cholesterol in his blood. Consequently, he is advised by a physician to stick to a low-fat diet under which high-fat consumption does not exceed the level of c. ¯ 5 Consuming high-fat products in excess of the recommended level bears an increased risk of undergoing a heart attack in the future. While the physician is unable, of course, to foresee the exact time that a heart attack will occur, he does have an opinion, which he shares with his patient, about the future distribution of the time of an attack. Suppose that F(t) represents the physician’s assessment of the probability that an attack will occur by some time t in the future, and F˙ (t)—his assessment of the probability that an attack will occur exactly at time t. Suppose further that the hazard rate, which is the probability of undergoing a heart attack at some time t in the future given that a heart attack has not occurred prior to that time, F˙ (t)/[1 − F (t)], is assessed by the physician to be an increasing, convex function of high-fat consumption and of a number of external risk factors, denoted by S, such as high blood pressure, diabetes, cigarette smoking, obesity, stress, personality type, genetic pre-disposition, etc. 6 A crucial question in modeling the individual’s problem is whether the hazard rate should be defined on current high-fat consumption or on the accumulated consumption rate. Because a heart attack is caused by the accumulation of cholesterol deposits on the artery walls, one first tends to relate the hazard rate at time t to the overall consumption of high-fat products until that time. This, however, implies that the risk of suffering an attack continues to increase (albeit at a lower rate) even if the individual restricts his high-fat consumption to c. ¯ Yet, recent evidence suggests that adhering to a low-fat diet reduces the 5 The American Heart Association has come up with two types of low-fat diets: a “step 1” diet, which is more moderate, and a “step 2”diet, for people with high cholesterol levels. The “step 2” diet restricts calories intake through saturated fat and total fat to 7 and 30% of the day’s total calories, respectively, and cholesterol intake to less than 200 mg a day, irrespective of total daily calories. As a comparison, the average American consumes 20 and 40% of total daily calories through saturated fat and total fat, respectively, and 500 mg of cholesterol daily (American Heart Association, 1991). 6 See Criqui (1986) for a review of the literature on risk factors contributing to CHD.
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risk of an attack, because it acts to dissolve the cholesterol deposits and widen the diameter of the arteries (!). 7 Defining the hazard rate as λ[c(t), S], where λc > 0 and λS > 0, thus, captures this phenomenon. 8 Consequently, the individual’s consumption in the past only affects the cumulative risk F(t) that a heart attack occurs by time t. If suffering a heart attack at some point of time t in the future, the individual is assumed to either die, with probability γ , or to receive life-saving treatment and completely recover. 9 Suppose that treatment costs are fully covered by health insurance and loss of income during recovery is fully compensated by sick-pay benefits, so that the only major harm caused to the individual if he does not die is the psychological shock accompanying the dreadful event and hospitalization in a coronary care unit, K. 10 Suppose, however, that the psychological shock is sufficiently intense to induce the individual to strictly adhere to the recommended diet, c, ¯ thereafter. While a risk for a second heart attack might still exist, there is nothing the individual can do to reduce it any further. We may, thus, simplify by assuming that adhering to the recommended diet ensures, irrespective of the level of S, that a second heart attack will not occur in the future. That is, λ(c, S) = 0 for c ≤ c. ¯ Suppose now that the individual must decide on whether or not to adhere to the prescribed diet prior to experiencing a heart attack, and if not—by how much to deviate from the physician’s advice, i.e. how much of high-fat products to consume in excess of the prescribed level. Suppose further that the individual decides on these questions through maximizing the present value of his expected lifetime utility stream from the consumption of high-fat and low-fat products, taking into account the adverse effect of high-fat consumption on the risk of a heart attack and its psychological and possibly deadly consequences. This may be viewed as a problem in optimal control, formulated as ∞ max e−δt {[1 − F (t)]U [c(t), h(t)] + F (t)(1 − γ )U¯ − F˙ (t)K} dt (1) 0
7 There are now at least nine published studies in which this possibility has been systematically examined and confirmed [see Pickering (1997) for details]. For example, in the St. Thomas’s Atheroma Regression Study (STARS), the progression of coronary artery narrowing was observed in 46% of persons belonging to a control group (who received no special treatment) but only in 12–15% of person in the treatment groups (whose blood cholesterol was lowered by diet, drugs, or both). An improvement in the diameter of the arteries was observed in only 4% of the controls, but in 38 and 33% of the treatment groups. Most importantly, the number of heart attacks and bypass operations appeared to be cut in half as a result of cholesterol-lowering treatment. 8 Even if cutting down high-fat consumption did not help dissolve cholesterol plaques, the important point in modeling non-adherence behavior is how people perceive the risk of a heart attack. Casual observation suggests that people believe (either because this is what doctors are telling them so as to induce them to keep a diet or because this is how they interpret doctors’ orders) that the risk of an attack can be drastically lowered through reducing current consumption of high-fat products. Defining the hazard rate on current consumption thus focuses on the really interesting question: if one believes that adhering to a low-fat diet lowers the risk of an attack, if not eliminates it altogether, why does he/she choose not to adhere? 9 The American Heart Association (1993) assesses that over 40% of the people who experience a heart attack in a given year die of it. Hence, γ , on average, is 0.4. In Section 3, the probability of dying if suffering a heart attack is made dependent on the delay in diagnosing an attack and obtaining emergency treatment. 10 Aside of having to face the fact that he has had a heart attack and the possibility of recurrence during the acute phase of hospitalization, a MI patient may experience cardiac arrest, having to be resuscitated through artificial means. This is likely to produce additional psychological problems such as nightmares, anxiety and depression (Druss and Korenfeld, 1967).
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subject to F˙ (t) = [1 − F (t)]λ[c(t), S],
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(2)
and h(t) = Y − c(t),
c(t) ≥ c, ¯
(3)
where δ denotes the discount rate of future utility, and U[c(t), h(t)] and U¯ ≡ U (c, ¯ Y − c) ¯ represent the individual’s pre-attack and post-attack utility levels from consumption at time t, respectively, assumed to be an increasing, concave function of both products, thus, satisfying Uc > 0, Uh > 0, Ucc < 0 and Uhh < 0. For simplicity, it is assumed that disposable income, Y, is constant over time and that both kind of products have the same price, regardless of their fat content, which is normalized to unity. The integrand (1) discounts the expected stream of lifetime utility over an infinite time horizon. 11 At any given time t in the future, the individual faces the cumulative probability 1−F (t) of not yet suffering a heart attack, deriving the utility U[c(t), h(t)] from consumption. However, with probability F(t), he will suffer a heart attack by this time, which, with probability γ , will be fatal, resulting in his death (the utility of which is assumed to be zero). Given the probability 1 − γ of surviving the event, the individual will thereafter, adhere to the recommended diet, deriving utility U¯ from consumption. In addition, with probability F˙ (t), a heart attack will occur exactly at time t, causing a psychological shock of size K. Notice that Eqs. (1)–(3) involve a state variable, F(t), which is subject to a dynamic constraint (2), and a control variable, c(t), which is subject to a static constraint (3). The control variable influences the objective function (1) directly (through its own value) and indirectly through its impact on the evolution of the state variable. Substituting Eqs. (2) and (3) into Eq. (1) and applying the Pontryagin Maximum Principle, the problem at hand involves the maximization of the Hamiltonian (omitting the time notation) H : (1 − F )[U (c, Y − c) − λ(c, S)K] + F (1 − γ )U¯ + q(1 − F )λ(c, S), (4) where q(t) is a dynamic multiplier representing the ‘shadow price’ of the cumulative probability of suffering an attack by time t. More precisely, q reflects the marginal ‘value’ to the individual of a slight increment to the overall risk, F. As long as an increase in F reduces expected utility, the accumulation of risk is undesirable and q will obtain a negative value. Maximization of Eq. (4) requires that the shadow price movement over time obeys the differential equation q˙ = −HF + δq = U (c, Y − c) − (1 − γ )U¯ − λ(c, S)K + [δ + λ(c, S)]q, (5) and that optimal high-fat consumption, c∗ , satisfies the first-order condition Hc = (1 − F )[Uc (c∗ , Y − c∗ ) − Uh (c∗ , Y − c∗ ) − λc (c∗ , S)(K − q)] = 0.
(6)
11 The simplifying assumption of an infinite planning horizon is quite common in dynamic health behavior models (e.g. Chaloupka, 1991; Francis, 1997). The infinite horizon model can easily be made more realistic by assuming a random date of death due to old age. The probability of surviving time t could be postulated as e−µt , which, incorporating into Eq. (1), would add the parameter µ to the discounting factor, δ. An individual with a higher level of µ would then discount the future more heavily than an individual with a lower µ (see Dardanoni, 1986).
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Condition (6) implies that excess fat consumption is independent of F, but dependent on its shadow price, q. Hence, the variation over time in the latter explains the variation over time in the former. However, it can be shown that the necessary conditions (5) and (6) are actually satisfied with constant values of c∗ and q, 12 which is essentially due to the assumption of an infinite planning horizon and to the exclusion of past consumption from the hazard rate. Setting now q˙ = 0 in Eq. (5), substituting into Eq. (6) and re-arranging yields MB ≡ Uc (c∗ , Y − c∗ ) − Uh (c∗ , Y − c∗ ) U (c∗ , Y − c∗ ) − (1 − γ )U¯ − λ(c∗ , S)K ∗ = λc (c , S) K + ≡ MC. δ + λ(c∗ , S)
(7)
Eq. (7) implies that high-fat consumption at each period of time preceding an attack should be determined such that the marginal benefit from deviating from the recommended diet (MB) equates the marginal cost (MC). The marginal benefit is reflected through the additional utility derived from consuming a unit of high-fat products rather than a unit of low-fat products. The marginal cost is reflected through the additional risk of suffering a heart attack emanating from consuming an additional unit of high-fat products. The increased risk involves not only the harm of suffering a psychological shock, but also the discounted value of the future utility loss dying from an attack as having to adhere, if surviving, to a low-fat diet, net of the expected psychological shock of an attack that might occur even if avoiding the additional unit of high-fat consumption. A sufficient condition for non-adherence is that at the level of the recommended diet, c, ¯ the marginal benefit from non-adherence exceeds the marginal cost. This will be the case if γ U¯ Uc (c, ¯ Y − c) ¯ − Uh (c, ¯ Y − c) ¯ > λc (c, ¯ S) K + . (8) δ Because λ(c, ¯ S) = 0, the marginal cost of non-adherence at c¯ reflects, in addition to the psychological shock, just the utility loss of future consumption if not surviving an attack, discounted by the individual’s time preference rate, δ. Clearly, an incentive for non-adherence is more likely to arise the lower the risk of suffering an attack due to a slight deviation from the recommended diet [λc (c, ¯ S)], the lower the psychological shock accompanying the dreadful event (K), the lower the probability of dying from an attack (γ ), and the lower the utility derived from adhering to the recommended diet if surviving an attack (U¯ ). An incentive for non-adherence is also more likely to arise the greater the individual’s rate of preference for the present (δ), and the greater his marginal “net” craving for high-fat products when adhering to the recommended diet [Uc (c, ¯ Y − c)−U ¯ ¯ Y − c)]. ¯ h (c, A pre-requisite for non-adherence is, of course, that the latter magnitude is positive. Given that condition (8) holds, the individual will opt to consume high-fat products in excess of the prescribed diet, raising the hazard rate to an above zero level. As non-adherence increases, the hazard rate will follow suit, shortening the expected time until a forthcoming attack. Consequently, the future must be discounted at a higher rate than the regular 12 For a detailed proof of the constancy over time of the optimal shadow price and control variable in a problem as such, see Kamien and Schwartz (1971a), who demonstrate this for a monopolist’s problem of setting a price which, if unrestrictively high, might attract the entry of rivals.
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time preference factor, which increases with the level of non-adherence. As is evident from Eq. (7), this acts to moderate the marginal cost of non-adherence, stimulating a greater consumption of high-fat products. 13 In sum, the hazard rate is not just a deterrent to non-adherence. It also imputes a lower value to future loss the greater the deviation from the recommended diet. This drives the individual to behave less respectfully towards his future, offering an economic explanation for a phenomenon traditionally attributed by psychologists to inability or reluctancy to change eating habits and lower life quality, or to senses of invulnerability (i.e. denying the risk of a heart attack) and fatalism (i.e. belief that nothing can help prevent a ‘destined’ attack). 14 As is immediately clear from inspecting Eq. (7), and is easily verified mathematically, 15 an increase in the probability of dying from an attack (γ ) or in the psychological shock accompanying an attack (K) would increase the marginal cost of non-adherence, thus, acting to discourage excess fat consumption. An increase in the marginal risk of suffering an attack [λc (c, S)] would have a similar effect. However, an increase in the hazard rate itself [λ(c, S)], holding its marginal rate constant, reduces the future loss component of the marginal cost, acting to stimulate excess fat consumption. Consequently, an increase in any of the external risk factors, S, which (assuming λcS ≥ 0) may affect both the hazard rate and its marginal rate, will have an ambiguous effect on the extent of deviation from the recommended diet. Interestingly, if λcS is zero or relatively small, an increase in an external risk factor will increase high-fat consumption, conforming with the fatalistic notion that if one believes that his time is short, he will seek to increase the quality of the time still left, adhering to the prophet’s complaint (“eat, drink, and be merry, for tomorrow we die”, Isaiah, 22:13) rather than to a low-fat diet. 16 While a psychological approach would attribute this behavior to 13 Notice that MC may also be written as λ {[U (c∗ , Y − c∗ ) − (1 − γ )U ¯ + δK]/(δ + λ)},which eliminates λ c from the numerator, thus, highlighting its role as a discount factor of future utility loss. 14 Recently, Becker and Mulligan (1997) presented a model of endogenous determination of time preference, arguing that “even rational people may ‘excessively’ discount future utilities” (p. 730). Becker and Mulligan focused, however, on the individual’s attempt to reduce the degree of overdiscounting, through spending resources on imagining (and improving the capacity to appreciate) future pleasures. 15 Totally differentiating Eq. (7) one obtains
λc U¯ dc∗ = ; dγ (λ + δ)∆
dc∗ λc δ = ; dK (λ + δ)∆
where ∆ ≡ Ucc − 2Uch − Uhh − λcc
dc∗ (λ + δ)λcS − λc λS = ; dS (λ + δ)2 ∆
U¯ c − U¯ h dc∗ = −λc (1 − γ ) , dc¯ λ + δ∆
U − (1 − γ )U¯ + δK <0 (λ + δ)
is the second-order condition for the maximization of Eq. (4). Assuming naturally that λcc ≥ 0, negativity of ∆ will be ensured if Uch ≥ 0. Otherwise, if Uch < 0, ∆ < 0 should be assumed to hold. 16 A possible risk factor not included in S could be the individual’s age (with the exception of age at the time of planning). In this case, the increased risk due to the progression of time could be captured through multiplying the hazard rate by a positive coefficient, a(t), where a (t) > 0. Kamien and Schwartz (1971b) used a somewhat similar formulation of the hazard rate to examine a problem in which preventive maintenance reduces the probability of machine failure, where the “natural” probability of failure increases with machine age. They show that the optimal maintenance rate for the machine is a non-increasing function of time. Analogizing “adherence” with “maintenance”, one may conclude that if the “natural” risk of a heart attack increases with age, optimal high-fat consumption would counter-intuitively be a non-decreasing function of time, supporting the result obtained above that non-adherence might actually increase with an increase in an external risk.
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a change in preferences towards present consumption, a rational economic analysis, which holds preferences stable, suggests that a greater hazard induces less adherence through discounting the future at a higher rate. Notice finally that the greater is c, ¯ the lower will be the future loss from an attack, since the less restrictive will be the post-attack diet. Hence, the greater is c, ¯ the lower will be the marginal cost of non-adherence, and the greater the pre-attack consumption of high-fat products. This suggests that a skillful physician may do better by prescribing a stricter than necessary diet, so as to induce a lower level of high-fat consumption. Such a strategy requires, however, that the physician has reasons to believe that the patient is not likely to adhere to the prescribed diet. Otherwise, if condition (8) does not hold, a stricter than necessary diet would have the opposite effect, driving the adhering patient to deviate from the prescribed diet.
3. Self-protection A major reason for the high mortality rates following heart attacks is the delay occurring in diagnosing an attack and obtaining emergency treatment. 17 Much of the blame falls on patients themselves, who often delay several hours or even days before seeking treatment. 18 Suppose, however, that the individual is fully aware of the crucial role of time in preventing a heart attack death, hence in case of suspecting an attack will apply for diagnosis and treatment promptly. He may also consider the possibility of self-protection against involuntary delay through measures that may help expedite diagnosis and treatment. 19 These may include subscribing to an emergency call-in center which provides round-the-clock cardiac diagnosis by phone, insuring with a modern intensive care ambulance service which adjusts its vehicle fleet and paramedics to the number of subscribers, or even relocating closer to a medical center. Formally, suppose that delay in diagnosis a heart attack and obtaining emergency treatment, D, determines the probability of dying in case of an attack, γ (D). More specifically, suppose that the probability of dying from an attack is an increasing, convex function of diagnosis and treatment delay, thus, γ (D) > 0 and γ (D) > 0. Suppose further that the ¯ and may opt to self-protect against individual assesses the public system delay to be D, the risk of dying from an attack through subscribing to private emergency services and/or 17 Many of the heart attack deaths are due to ventricular fibrillation (i.e. chaotic electrical disturbances) of the heart that occurs before the victim can reach any medical assistance or an emergency room. When paramedics arrive, medication and/or electrical shock to the heart can be promptly administered to convert ventricular fibrillation to a normal heart rhythm. Once in an emergency room, electrical disturbances are treated through administration of medications to dissolve blood clots and open blood vessels, and/or balloon angioplasty to open an obstructed artery. Therefore, 90–95% of heart attack victims who make it to the hospital survive (American Heart Association, 1993). 18 Some patients are unaware of the importance of prompt diagnosis, and others are simply unable to face the fact that they have had a heart attack, interpreting the symptoms as mild disorders. See Taylor (1995) for a discussion of delay behavior in MI episodes. 19 Self-protection, defined as measures taken to reduce the probability of a loss, was first analyzed in the literature by Ehrlich and Becker (1972), as an alternative to market insurance and to self-insurance which reduce the size of a loss.
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expending on other types of delay-reducing measures. Total expenditure, ϕ(D), is assumed to be a decreasing, convex function of delay, thus, ϕ (D) < 0 and ϕ (D) > 0, where ¯ = 0. That is, the shorter the desired delay, the greater the expenditure (which inϕ(D) creases at increasing marginal rates). Allowing the individual to determine both the desired levels of delay, D∗ , and high-fat consumption, c∗ , his problem becomes t ∞ −δt ¯ ˙ ˙ [1 − F (t)]U [c(t), h(t)] + U F (k)[1 − γ (D(k))] dk − F (t)K dt e max 0
0
(9) subject to F˙ (t) = [1 − F (t)]λ[c(t), S],
(10)
and h(t) = Y − c(t) − ϕ[D(t)],
¯ c(t) ≥ c, ¯ D(t) ≤ D.
(11)
Notice that if an attack has already occurred by time t, expected utility takes account of the possibility that the attack could have occurred in any particular time in the past, thus, weighting the probability of suffering an attack at any point of time in the past by the probability of surviving as determined by the level of delay existing at that time. Notice also that if an attack has already occurred by time t, the individual will not expend on delay-reducing measures at that time, since adhering to the prescribed diet ensures, by assumption, that a second attack will not occur in the future. Substituting Eqs. (10) and (11) into Eq. (9), the Hamiltonian to be maximized is H : (1 − F ){U [c, Y − c − ϕ(D)] − λ(c, S)K} t ¯ + U [1 − F (k)]λ[c(k), S][1 − γ (D(k))] dk + q(1 − F )λ(c, S).
(12)
0
The necessary condition for the evolution of the shadow price over time now becomes q˙ = U (c, h) − λ(c, S)[K − (1 − γ (D))U¯ ] + [δ + λ(c, S)]q,
(13)
whereas the necessary first-order conditions determining c∗ and D∗ are given by Hc = (1 − F ){[Uc (c, h) − Uh (c, h)] − λc (c, S)[K − q − (1 − γ (D))U¯ ]} = 0 (14) HD = (1 − F )[−ϕ (D)Uh (c, h) − λ(c, S)γ (D)U¯ ] = 0.
(15)
A sufficient condition for self-protection, at any given c, is that the marginal benefit from ¯ exceeds the marginal cost of doing so. reducing delay below the public system level D, ¯ this requires that Evaluating condition (15) at D, ¯ U¯ > −ϕ (D)U ¯ h (c, Y − c). λ(c, S)γ (D)
(16)
Evidently, an incentive for self-protection will arise if the marginal expenditure on reduc¯ is sufficiently small. An incentive for self-protection is also more likely ing delay [−ϕ (D)] to arise the greater the risk of suffering an attack [λ(c, S)], the greater the marginal reduction
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¯ or the greater the utility level if surviving an in the risk of dying from an attack [γ (D)], attack (U¯ ). How would high-fat consumption respond to the opportunity of self-protecting against the risk of dying from an attack? To examine this question, suppose, for simplicity, that ¯ ϕ(D) = π(D−D), where π[= −ϕ (D)] denotes the cost of reducing delay by an additional unit of time. Suppose also that π is just high enough so that the individual has no incentive to self-protect (i.e. condition (16) holds as an equality), and consider the effect of a fall in π on D∗ and c∗ . Setting first q˙ = 0 in Eq. (13) and substituting into Eq. (14), yields Hc = Uc (c, h) − Uh (c, h) − λc (c, S) U (c, h) − δ(1 − γ (D))U¯ − λ(c, S)K × K+ = 0. δ + λ(c, S)
(14 )
Totally differentiating Eqs. (14 ) and (15) with respect to D, c, and π , solving for dD∗ /dπ ¯ we obtain and dc∗ /dπ , and evaluating the results at D = D, dD ∗ −Hcc Uh (c, h) ¯ = 2 dπ D=D Hcc HDD − HcD
(17)
HcD Uh (c, h) dc∗ , ¯ = 2 dπ D=D Hcc HDD − HcD
(18)
where the second-order conditions for the maximization of the Hamiltonian require that 2 > 0. While the sign of Eq. (17) is unambiguously positive, imHcc < 0 and Hcc HDD −HcD ¯ the sign of Eq. (18) depends plying that a fall in π at D¯ will induce self-protection (D ∗ ≤ D), ¯ U¯ . 20 on the sign of HcD , which is ambiguous, since HcD = HDc = π(Uch −Uhh )−λc γ (D) Hence, if HcD is negative, self-protection will be accompanied with an increased level of high-fat consumption (therefore, less adherence to the prescribed diet), whereas if HcD is positive, self-protection will be accompanied with a lowered level of high-fat consumption (therefore, more adherence to the prescribed diet). Quite interestingly, and contrary to prior intuition, subscribing to a self-protection scheme must not necessarily give rise to a “moral hazard” effect in the form of stimulating health negligence behavior. That is, dietary adherence and self-protection may be complements in the sense that a fall in price, which induces the latter, enhances the former. It thus follows that public health policy, informed of private preferences, might be able to reduce both the risk of a heart attack and the risk of dying from an attack through subsidizing the price of private emergency services. Rather than trying to change people, public policy could try changing the costs they face.
4. Concluding remarks The present paper has applied a rational expected–utility–maximization model to inquire into the individual’s decision to deviate from a prescribed low-fat diet, despite bearing 20 This may be obtained either through differentiating Eq. (15) with respect to c, or differentiating Eq. (14 ) with respect to D, and substituting Eq. (15).
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an increased risk of suffering a heart attack in the future. The analysis has given rise to a number of interesting conclusions regarding dietary non-adherence and public health policy. First, the risk of undergoing a heart attack assumes a double-agent role in the rational decision model: while acting to deter high-fat consumption, it also enhances non-adherence through discounting future costs at a higher rate, which is positively dependent on the extent of deviation from the recommended diet. In its latter role as a discount factor which increases with the level of non-adherence, the risk of an attack drives the individual to behave less respectfully towards the future. Second, the greater the level of external risk factors, such as smoking, diabetes, obesity, or genetic pre-disposition, the greater may be excess fat consumption, as the shorter the expected time left for the individual to enjoy life. Third, a skillful physician may induce lower consumption of high-fat products through prescribing a stricter than necessary diet. Fourth, contrary to intuition, dietary adherence and self-protection to reduce delay in diagnosis and treatment of an attack can be complements, in the sense that engaging in the latter enhances the former. Public health policy may, thus, help reduce both the risk of an attack and the risk of dying from an attack through providing incentives for high-risk individuals to subscribe to delay-reducing emergency services. The model developed in this paper may be extended in several ways, although at the cost of substantial mathematical complexity. First, rather than assuming that the individual learns his lesson after the first attack, adhering thereafter, to the recommended diet, the possibility of post-attack non-adherence, at the risk of suffering a second attack, may be allowed for. It would be interesting to see whether optimal post-attack adherence is indeed greater than its pre-attack counterpart, or whether it may be rational for the individual to exhibit even a greater disrespect towards his future. Secondly, the possibility of permanent physical and emotional impairments following an attack may be incorporated into the model, rather than restricting its consequences to the extreme cases of death and complete recovery. This could make the model relevant to the case of brain stroke as well, the risk factors for which overlap heavily with those for a heart attack, but which is more likely to result in significant neurological and cognitive impairments. Finally, self-protection measures which reduce not just the risk of dying from an attack, but also the risk of an attack itself, such as cholesterol-lowering drugs (at the cost of unpleasant side effects) or aerobic exercising (at the expense of leisure time), may be incorporated into the model, contributing to a more comprehensive understanding of the non-adherence phenomenon.
Acknowledgements I am indebted to Amir Shmueli and two anonymous referees for helpful comments and suggestions. References American Heart Association, 1991. Low-fat, Low Cholesterol Cook Book. Times Books, New York. American Heart Association, 1993. Heart and Stroke Facts. Dallas, TX. Becker, G.S., Mulligan, C.B., 1997. The endogenous determination of time preference. Quarterly Journal of Economics 112, 729–758.
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Becker, G.S., Murphy, K.M., 1988. A theory of rational addiction. Journal of Political Economy 98, 675–700. Carmody, T.P., Matarazzo, J.D., Istvan, J.A., 1987. Promoting adherence to heart-healthy diets: a review of the literature. Journal of Compliance in Health Care 2, 105–124. Chaloupka, F., 1991. Rational addictive behavior and cigarette smoking. Journal of Political Economy 99, 722–742. Criqui, M.H., 1986. Epidemiology of atherosclerosis: an updated overview. American Journal of Cardiology 57, 18C–23C. Dardanoni, V., 1986. A note on a simple model of health investment. Bulletin of Economic Research 38, 97–100. Druss, R.G., Korenfeld, D.S., 1967. Survivors of cardiac arrest: psychiatric study. Journal of American Medical Association 201, 291–296. Ehrlich, I., Becker, G.S., 1972. Market insurance, self-insurance, and self-protection. Journal of Political Economy 80, 623–648. Francis, P.J., 1997. Dynamic epidemiology and the market for vaccination. Journal of Public Economics 63, 383–406. Hamermesh, D.S., Soss, N.M., 1974. An economic theory of suicide. Journal of Political Economy 82, 83–98. Kamien, M.I., Schwartz, N.L., 1971a. Limit pricing and uncertain entry. Econometrica 39, 441–454. Kamien, M.I., Schwartz, N.L., 1971b. Optimal maintenance and sale age for a machine subject to failure. Management Science 17 ((B)), 495–504. Pickering, T., 1997. Good News About High Blood Pressure. Simon & Schuster, New York. Taylor, S.E., 1995. Health Psychology. McGraw-Hill, Singapore. Winston, G.C., 1980. Addiction and backsliding: a theory of compulsive consumption. Journal of Economic Behavior and Organization 1, 295–324. Yaniv, G., 1998. Phobic disorder, psychotherapy, and risk-taking: an economic perspective. Journal of Health Economics 17, 229–244.
Non-adherence to a low-fat diet: an economic perspective Gideon Yaniv a,b,∗ a
National Insurance Institute, 13 Weitzman Avenue, Jerusalem 95437, Israel b College of Management, Tel Aviv, Israel Received 15 November 1999; accepted 16 October 2000
Abstract Despite bearing increased risk of suffering a heart attack, failure to adhere to a low-fat dietary regimen is wide-spread. The paper applies an expected–utility–maximization model to inquire into the individual’s decision to deviate from a prescribed low-fat diet, offering an economic explanation for excess high-fat consumption. The implications of self-protection against the risk of dying from an attack are further examined. The analysis reveals that self-protection and dietary adherence can be complements, suggesting that public health policy may help reduce both the risk of an attack and the risk of dying from an attack through providing incentives for self-protection. © 2002 Elsevier Science B.V. All rights reserved. JEL classification: I12; I18; D81 PsycINFO classification: 3360; 3361 Keywords: Low-fat diet; Non-adherence; Heart attack; Hazard rate; Self-protection
1. Introduction Coronary heart disease (CHD), caused by the narrowing of the coronary arteries that supply blood to the heart, is the number one killer in the US, accounting for 40% of all deaths. It is considered, in part, to be a disease of lifestyle, associated with a number of poor health habits, such as a high-fat, cholesterol-rich diet, smoking and a low activity level. A primary clinical manifestation of CHD is myocardial infarction (MI), otherwise known
∗ Tel.: +972-2-6709561; fax: +972-2-6528508. E-mail address: [email protected] (G. Yaniv).
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as a heart attack, which is the death of the heart muscle due to the loss of blood supply. 1 Approximately, 1 million Americans suffer a heart attack annually, 400,000 die as a result. Consequently, modification of CHD risk-related behavior to reduce the incidence of CHD morbidity and mortality obtain high priority in public health intervention efforts. 2 However, getting people to develop good health habits and give up the risky behaviors that damage their health is not an easy task. In particular, despite convincing evidence that adherence to low-fat diet can significantly lower blood cholesterol levels, 3 failure to adhere to dietary restrictions is wide-spread. Why is it then that high-risk individuals might not adhere to prescribed dietary regimens? Health psychology suggests a number of explanations: 4 preferences for meat and fat dairy products are well established and hard to alter; dietary recommendations might necessitate drastic changes in meal planning, cooking methods and eating habits; they may be too restrictive, monotonous, and associated with a lower quality of life. In addition, a low sense of self-efficacy (i.e. disbelief in one’s ability to change one’s diet), or a strong sense of invulnerability or fatalism may attenuate adherence. The present paper addresses this question from an economic standpoint, proposing a rational decision-making model to explain individuals’ deviation from a prescribed low-fat diet. Most importantly, while psychological explanations focus on habit, reduced life quality, or irrational beliefs as deterrents to adherence, the economic model takes explicit account of the risk involved, confronting future costs with present benefits of non-adherence. The analysis reveals that the risk of undergoing a heart attack, which is assumed to increase with the level of excess fat consumption, is not just a deterrent to non-adherence. It also plays the role of a discount factor of future costs, reducing their value more the greater the deviation from the recommended diet. This drives the individual to behave less respectfully towards the future, offering an economic motivation for a seemingly habitual or irrational behavior. Another interesting result is that an increase in the level of external risk factors (e.g. high blood pressure, diabetes, stress, family history), might increase high-fat consumption, conforming with the fatalistic notion that if one believes that his time is short, he will seek increased enjoyment from the time still left. The present paper belongs to as yet a small literature which applies the rational economic approach to analyzing traditional health/clinical psychology phenomena, such as compulsive consumption (Winston, 1980), addictive behavior (Becker and Murphy, 1988), suicide threat (Hamermesh and Soss, 1974), or fear of public places (Yaniv, 1998). The paper begins (Section 2) with modeling the individual’s decision to consume high-fat products in 1 A heart attack most often occurs due to the accumulation of cholesterol (a fatty chemical found mainly in animal products) on the inner walls of an artery to the heart. Over time, the build-up of cholesterol deposits may cause substantial thickening of the artery walls and significant narrowing of the artery. Occasionally, the cholesterol plaque becomes sticky, triggering a blood clot to form on its surface that completely blocks the artery (American Heart Association, 1993). 2 In 1985, the National Institute of Health (NIH) launched a national campaign known as the National Cholesterol Education Program (NCEP), with the purpose of reducing illness and deaths from CHD through encouraging the general public to go on a low-fat diet. Recent surveys show that from 1985 to 1995, the percentage of Americans who had ever had their blood cholesterol checked rose from 35 to 75 and that blood cholesterol levels among US adults dropped, on average, from 213 to 203 mg/dl. 3 See Carmody et al. (1987) for a review of the literature. 4 See Taylor (1995, Chapter 5) for a detailed discussion of the psychological explanations and related literature.
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excess of a prescribed low-fat diet, risking the occurrence of a heart attack at some uncertain time in the future, the distribution of which is affected by excess fat consumption. It then proceeds (Section 3) with examining the implications of self-protection against the risk of dying from an attack, through subscribing to emergency services that reduce delay in diagnosing an attack or in obtaining emergency treatment. The analysis reveals, counter-intuitively, that self-protection and dietary adherence can be complements, suggesting that public health policy may help reduce both the risk of an attack and the risk of dying from an attack through providing incentives for subscribing to delay-reducing services. The paper concludes (Section 4) with a summary of the main results and some related remarks.
2. The model Consider an individual, who, during a given period of time, spends his disposable income, Y, on the consumption of high-fat, cholesterol-rich products, c, and low-fat, cholesterol-free products, h. Suppose that a blood test taken at the beginning of the period reveals abovenormal values of cholesterol in his blood. Consequently, he is advised by a physician to stick to a low-fat diet under which high-fat consumption does not exceed the level of c. ¯ 5 Consuming high-fat products in excess of the recommended level bears an increased risk of undergoing a heart attack in the future. While the physician is unable, of course, to foresee the exact time that a heart attack will occur, he does have an opinion, which he shares with his patient, about the future distribution of the time of an attack. Suppose that F(t) represents the physician’s assessment of the probability that an attack will occur by some time t in the future, and F˙ (t)—his assessment of the probability that an attack will occur exactly at time t. Suppose further that the hazard rate, which is the probability of undergoing a heart attack at some time t in the future given that a heart attack has not occurred prior to that time, F˙ (t)/[1 − F (t)], is assessed by the physician to be an increasing, convex function of high-fat consumption and of a number of external risk factors, denoted by S, such as high blood pressure, diabetes, cigarette smoking, obesity, stress, personality type, genetic pre-disposition, etc. 6 A crucial question in modeling the individual’s problem is whether the hazard rate should be defined on current high-fat consumption or on the accumulated consumption rate. Because a heart attack is caused by the accumulation of cholesterol deposits on the artery walls, one first tends to relate the hazard rate at time t to the overall consumption of high-fat products until that time. This, however, implies that the risk of suffering an attack continues to increase (albeit at a lower rate) even if the individual restricts his high-fat consumption to c. ¯ Yet, recent evidence suggests that adhering to a low-fat diet reduces the 5 The American Heart Association has come up with two types of low-fat diets: a “step 1” diet, which is more moderate, and a “step 2”diet, for people with high cholesterol levels. The “step 2” diet restricts calories intake through saturated fat and total fat to 7 and 30% of the day’s total calories, respectively, and cholesterol intake to less than 200 mg a day, irrespective of total daily calories. As a comparison, the average American consumes 20 and 40% of total daily calories through saturated fat and total fat, respectively, and 500 mg of cholesterol daily (American Heart Association, 1991). 6 See Criqui (1986) for a review of the literature on risk factors contributing to CHD.
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risk of an attack, because it acts to dissolve the cholesterol deposits and widen the diameter of the arteries (!). 7 Defining the hazard rate as λ[c(t), S], where λc > 0 and λS > 0, thus, captures this phenomenon. 8 Consequently, the individual’s consumption in the past only affects the cumulative risk F(t) that a heart attack occurs by time t. If suffering a heart attack at some point of time t in the future, the individual is assumed to either die, with probability γ , or to receive life-saving treatment and completely recover. 9 Suppose that treatment costs are fully covered by health insurance and loss of income during recovery is fully compensated by sick-pay benefits, so that the only major harm caused to the individual if he does not die is the psychological shock accompanying the dreadful event and hospitalization in a coronary care unit, K. 10 Suppose, however, that the psychological shock is sufficiently intense to induce the individual to strictly adhere to the recommended diet, c, ¯ thereafter. While a risk for a second heart attack might still exist, there is nothing the individual can do to reduce it any further. We may, thus, simplify by assuming that adhering to the recommended diet ensures, irrespective of the level of S, that a second heart attack will not occur in the future. That is, λ(c, S) = 0 for c ≤ c. ¯ Suppose now that the individual must decide on whether or not to adhere to the prescribed diet prior to experiencing a heart attack, and if not—by how much to deviate from the physician’s advice, i.e. how much of high-fat products to consume in excess of the prescribed level. Suppose further that the individual decides on these questions through maximizing the present value of his expected lifetime utility stream from the consumption of high-fat and low-fat products, taking into account the adverse effect of high-fat consumption on the risk of a heart attack and its psychological and possibly deadly consequences. This may be viewed as a problem in optimal control, formulated as ∞ max e−δt {[1 − F (t)]U [c(t), h(t)] + F (t)(1 − γ )U¯ − F˙ (t)K} dt (1) 0
7 There are now at least nine published studies in which this possibility has been systematically examined and confirmed [see Pickering (1997) for details]. For example, in the St. Thomas’s Atheroma Regression Study (STARS), the progression of coronary artery narrowing was observed in 46% of persons belonging to a control group (who received no special treatment) but only in 12–15% of person in the treatment groups (whose blood cholesterol was lowered by diet, drugs, or both). An improvement in the diameter of the arteries was observed in only 4% of the controls, but in 38 and 33% of the treatment groups. Most importantly, the number of heart attacks and bypass operations appeared to be cut in half as a result of cholesterol-lowering treatment. 8 Even if cutting down high-fat consumption did not help dissolve cholesterol plaques, the important point in modeling non-adherence behavior is how people perceive the risk of a heart attack. Casual observation suggests that people believe (either because this is what doctors are telling them so as to induce them to keep a diet or because this is how they interpret doctors’ orders) that the risk of an attack can be drastically lowered through reducing current consumption of high-fat products. Defining the hazard rate on current consumption thus focuses on the really interesting question: if one believes that adhering to a low-fat diet lowers the risk of an attack, if not eliminates it altogether, why does he/she choose not to adhere? 9 The American Heart Association (1993) assesses that over 40% of the people who experience a heart attack in a given year die of it. Hence, γ , on average, is 0.4. In Section 3, the probability of dying if suffering a heart attack is made dependent on the delay in diagnosing an attack and obtaining emergency treatment. 10 Aside of having to face the fact that he has had a heart attack and the possibility of recurrence during the acute phase of hospitalization, a MI patient may experience cardiac arrest, having to be resuscitated through artificial means. This is likely to produce additional psychological problems such as nightmares, anxiety and depression (Druss and Korenfeld, 1967).
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subject to F˙ (t) = [1 − F (t)]λ[c(t), S],
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(2)
and h(t) = Y − c(t),
c(t) ≥ c, ¯
(3)
where δ denotes the discount rate of future utility, and U[c(t), h(t)] and U¯ ≡ U (c, ¯ Y − c) ¯ represent the individual’s pre-attack and post-attack utility levels from consumption at time t, respectively, assumed to be an increasing, concave function of both products, thus, satisfying Uc > 0, Uh > 0, Ucc < 0 and Uhh < 0. For simplicity, it is assumed that disposable income, Y, is constant over time and that both kind of products have the same price, regardless of their fat content, which is normalized to unity. The integrand (1) discounts the expected stream of lifetime utility over an infinite time horizon. 11 At any given time t in the future, the individual faces the cumulative probability 1−F (t) of not yet suffering a heart attack, deriving the utility U[c(t), h(t)] from consumption. However, with probability F(t), he will suffer a heart attack by this time, which, with probability γ , will be fatal, resulting in his death (the utility of which is assumed to be zero). Given the probability 1 − γ of surviving the event, the individual will thereafter, adhere to the recommended diet, deriving utility U¯ from consumption. In addition, with probability F˙ (t), a heart attack will occur exactly at time t, causing a psychological shock of size K. Notice that Eqs. (1)–(3) involve a state variable, F(t), which is subject to a dynamic constraint (2), and a control variable, c(t), which is subject to a static constraint (3). The control variable influences the objective function (1) directly (through its own value) and indirectly through its impact on the evolution of the state variable. Substituting Eqs. (2) and (3) into Eq. (1) and applying the Pontryagin Maximum Principle, the problem at hand involves the maximization of the Hamiltonian (omitting the time notation) H : (1 − F )[U (c, Y − c) − λ(c, S)K] + F (1 − γ )U¯ + q(1 − F )λ(c, S), (4) where q(t) is a dynamic multiplier representing the ‘shadow price’ of the cumulative probability of suffering an attack by time t. More precisely, q reflects the marginal ‘value’ to the individual of a slight increment to the overall risk, F. As long as an increase in F reduces expected utility, the accumulation of risk is undesirable and q will obtain a negative value. Maximization of Eq. (4) requires that the shadow price movement over time obeys the differential equation q˙ = −HF + δq = U (c, Y − c) − (1 − γ )U¯ − λ(c, S)K + [δ + λ(c, S)]q, (5) and that optimal high-fat consumption, c∗ , satisfies the first-order condition Hc = (1 − F )[Uc (c∗ , Y − c∗ ) − Uh (c∗ , Y − c∗ ) − λc (c∗ , S)(K − q)] = 0.
(6)
11 The simplifying assumption of an infinite planning horizon is quite common in dynamic health behavior models (e.g. Chaloupka, 1991; Francis, 1997). The infinite horizon model can easily be made more realistic by assuming a random date of death due to old age. The probability of surviving time t could be postulated as e−µt , which, incorporating into Eq. (1), would add the parameter µ to the discounting factor, δ. An individual with a higher level of µ would then discount the future more heavily than an individual with a lower µ (see Dardanoni, 1986).
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Condition (6) implies that excess fat consumption is independent of F, but dependent on its shadow price, q. Hence, the variation over time in the latter explains the variation over time in the former. However, it can be shown that the necessary conditions (5) and (6) are actually satisfied with constant values of c∗ and q, 12 which is essentially due to the assumption of an infinite planning horizon and to the exclusion of past consumption from the hazard rate. Setting now q˙ = 0 in Eq. (5), substituting into Eq. (6) and re-arranging yields MB ≡ Uc (c∗ , Y − c∗ ) − Uh (c∗ , Y − c∗ ) U (c∗ , Y − c∗ ) − (1 − γ )U¯ − λ(c∗ , S)K ∗ = λc (c , S) K + ≡ MC. δ + λ(c∗ , S)
(7)
Eq. (7) implies that high-fat consumption at each period of time preceding an attack should be determined such that the marginal benefit from deviating from the recommended diet (MB) equates the marginal cost (MC). The marginal benefit is reflected through the additional utility derived from consuming a unit of high-fat products rather than a unit of low-fat products. The marginal cost is reflected through the additional risk of suffering a heart attack emanating from consuming an additional unit of high-fat products. The increased risk involves not only the harm of suffering a psychological shock, but also the discounted value of the future utility loss dying from an attack as having to adhere, if surviving, to a low-fat diet, net of the expected psychological shock of an attack that might occur even if avoiding the additional unit of high-fat consumption. A sufficient condition for non-adherence is that at the level of the recommended diet, c, ¯ the marginal benefit from non-adherence exceeds the marginal cost. This will be the case if γ U¯ Uc (c, ¯ Y − c) ¯ − Uh (c, ¯ Y − c) ¯ > λc (c, ¯ S) K + . (8) δ Because λ(c, ¯ S) = 0, the marginal cost of non-adherence at c¯ reflects, in addition to the psychological shock, just the utility loss of future consumption if not surviving an attack, discounted by the individual’s time preference rate, δ. Clearly, an incentive for non-adherence is more likely to arise the lower the risk of suffering an attack due to a slight deviation from the recommended diet [λc (c, ¯ S)], the lower the psychological shock accompanying the dreadful event (K), the lower the probability of dying from an attack (γ ), and the lower the utility derived from adhering to the recommended diet if surviving an attack (U¯ ). An incentive for non-adherence is also more likely to arise the greater the individual’s rate of preference for the present (δ), and the greater his marginal “net” craving for high-fat products when adhering to the recommended diet [Uc (c, ¯ Y − c)−U ¯ ¯ Y − c)]. ¯ h (c, A pre-requisite for non-adherence is, of course, that the latter magnitude is positive. Given that condition (8) holds, the individual will opt to consume high-fat products in excess of the prescribed diet, raising the hazard rate to an above zero level. As non-adherence increases, the hazard rate will follow suit, shortening the expected time until a forthcoming attack. Consequently, the future must be discounted at a higher rate than the regular 12 For a detailed proof of the constancy over time of the optimal shadow price and control variable in a problem as such, see Kamien and Schwartz (1971a), who demonstrate this for a monopolist’s problem of setting a price which, if unrestrictively high, might attract the entry of rivals.
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time preference factor, which increases with the level of non-adherence. As is evident from Eq. (7), this acts to moderate the marginal cost of non-adherence, stimulating a greater consumption of high-fat products. 13 In sum, the hazard rate is not just a deterrent to non-adherence. It also imputes a lower value to future loss the greater the deviation from the recommended diet. This drives the individual to behave less respectfully towards his future, offering an economic explanation for a phenomenon traditionally attributed by psychologists to inability or reluctancy to change eating habits and lower life quality, or to senses of invulnerability (i.e. denying the risk of a heart attack) and fatalism (i.e. belief that nothing can help prevent a ‘destined’ attack). 14 As is immediately clear from inspecting Eq. (7), and is easily verified mathematically, 15 an increase in the probability of dying from an attack (γ ) or in the psychological shock accompanying an attack (K) would increase the marginal cost of non-adherence, thus, acting to discourage excess fat consumption. An increase in the marginal risk of suffering an attack [λc (c, S)] would have a similar effect. However, an increase in the hazard rate itself [λ(c, S)], holding its marginal rate constant, reduces the future loss component of the marginal cost, acting to stimulate excess fat consumption. Consequently, an increase in any of the external risk factors, S, which (assuming λcS ≥ 0) may affect both the hazard rate and its marginal rate, will have an ambiguous effect on the extent of deviation from the recommended diet. Interestingly, if λcS is zero or relatively small, an increase in an external risk factor will increase high-fat consumption, conforming with the fatalistic notion that if one believes that his time is short, he will seek to increase the quality of the time still left, adhering to the prophet’s complaint (“eat, drink, and be merry, for tomorrow we die”, Isaiah, 22:13) rather than to a low-fat diet. 16 While a psychological approach would attribute this behavior to 13 Notice that MC may also be written as λ {[U (c∗ , Y − c∗ ) − (1 − γ )U ¯ + δK]/(δ + λ)},which eliminates λ c from the numerator, thus, highlighting its role as a discount factor of future utility loss. 14 Recently, Becker and Mulligan (1997) presented a model of endogenous determination of time preference, arguing that “even rational people may ‘excessively’ discount future utilities” (p. 730). Becker and Mulligan focused, however, on the individual’s attempt to reduce the degree of overdiscounting, through spending resources on imagining (and improving the capacity to appreciate) future pleasures. 15 Totally differentiating Eq. (7) one obtains
λc U¯ dc∗ = ; dγ (λ + δ)∆
dc∗ λc δ = ; dK (λ + δ)∆
where ∆ ≡ Ucc − 2Uch − Uhh − λcc
dc∗ (λ + δ)λcS − λc λS = ; dS (λ + δ)2 ∆
U¯ c − U¯ h dc∗ = −λc (1 − γ ) , dc¯ λ + δ∆
U − (1 − γ )U¯ + δK <0 (λ + δ)
is the second-order condition for the maximization of Eq. (4). Assuming naturally that λcc ≥ 0, negativity of ∆ will be ensured if Uch ≥ 0. Otherwise, if Uch < 0, ∆ < 0 should be assumed to hold. 16 A possible risk factor not included in S could be the individual’s age (with the exception of age at the time of planning). In this case, the increased risk due to the progression of time could be captured through multiplying the hazard rate by a positive coefficient, a(t), where a (t) > 0. Kamien and Schwartz (1971b) used a somewhat similar formulation of the hazard rate to examine a problem in which preventive maintenance reduces the probability of machine failure, where the “natural” probability of failure increases with machine age. They show that the optimal maintenance rate for the machine is a non-increasing function of time. Analogizing “adherence” with “maintenance”, one may conclude that if the “natural” risk of a heart attack increases with age, optimal high-fat consumption would counter-intuitively be a non-decreasing function of time, supporting the result obtained above that non-adherence might actually increase with an increase in an external risk.
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a change in preferences towards present consumption, a rational economic analysis, which holds preferences stable, suggests that a greater hazard induces less adherence through discounting the future at a higher rate. Notice finally that the greater is c, ¯ the lower will be the future loss from an attack, since the less restrictive will be the post-attack diet. Hence, the greater is c, ¯ the lower will be the marginal cost of non-adherence, and the greater the pre-attack consumption of high-fat products. This suggests that a skillful physician may do better by prescribing a stricter than necessary diet, so as to induce a lower level of high-fat consumption. Such a strategy requires, however, that the physician has reasons to believe that the patient is not likely to adhere to the prescribed diet. Otherwise, if condition (8) does not hold, a stricter than necessary diet would have the opposite effect, driving the adhering patient to deviate from the prescribed diet.
3. Self-protection A major reason for the high mortality rates following heart attacks is the delay occurring in diagnosing an attack and obtaining emergency treatment. 17 Much of the blame falls on patients themselves, who often delay several hours or even days before seeking treatment. 18 Suppose, however, that the individual is fully aware of the crucial role of time in preventing a heart attack death, hence in case of suspecting an attack will apply for diagnosis and treatment promptly. He may also consider the possibility of self-protection against involuntary delay through measures that may help expedite diagnosis and treatment. 19 These may include subscribing to an emergency call-in center which provides round-the-clock cardiac diagnosis by phone, insuring with a modern intensive care ambulance service which adjusts its vehicle fleet and paramedics to the number of subscribers, or even relocating closer to a medical center. Formally, suppose that delay in diagnosis a heart attack and obtaining emergency treatment, D, determines the probability of dying in case of an attack, γ (D). More specifically, suppose that the probability of dying from an attack is an increasing, convex function of diagnosis and treatment delay, thus, γ (D) > 0 and γ (D) > 0. Suppose further that the ¯ and may opt to self-protect against individual assesses the public system delay to be D, the risk of dying from an attack through subscribing to private emergency services and/or 17 Many of the heart attack deaths are due to ventricular fibrillation (i.e. chaotic electrical disturbances) of the heart that occurs before the victim can reach any medical assistance or an emergency room. When paramedics arrive, medication and/or electrical shock to the heart can be promptly administered to convert ventricular fibrillation to a normal heart rhythm. Once in an emergency room, electrical disturbances are treated through administration of medications to dissolve blood clots and open blood vessels, and/or balloon angioplasty to open an obstructed artery. Therefore, 90–95% of heart attack victims who make it to the hospital survive (American Heart Association, 1993). 18 Some patients are unaware of the importance of prompt diagnosis, and others are simply unable to face the fact that they have had a heart attack, interpreting the symptoms as mild disorders. See Taylor (1995) for a discussion of delay behavior in MI episodes. 19 Self-protection, defined as measures taken to reduce the probability of a loss, was first analyzed in the literature by Ehrlich and Becker (1972), as an alternative to market insurance and to self-insurance which reduce the size of a loss.
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expending on other types of delay-reducing measures. Total expenditure, ϕ(D), is assumed to be a decreasing, convex function of delay, thus, ϕ (D) < 0 and ϕ (D) > 0, where ¯ = 0. That is, the shorter the desired delay, the greater the expenditure (which inϕ(D) creases at increasing marginal rates). Allowing the individual to determine both the desired levels of delay, D∗ , and high-fat consumption, c∗ , his problem becomes t ∞ −δt ¯ ˙ ˙ [1 − F (t)]U [c(t), h(t)] + U F (k)[1 − γ (D(k))] dk − F (t)K dt e max 0
0
(9) subject to F˙ (t) = [1 − F (t)]λ[c(t), S],
(10)
and h(t) = Y − c(t) − ϕ[D(t)],
¯ c(t) ≥ c, ¯ D(t) ≤ D.
(11)
Notice that if an attack has already occurred by time t, expected utility takes account of the possibility that the attack could have occurred in any particular time in the past, thus, weighting the probability of suffering an attack at any point of time in the past by the probability of surviving as determined by the level of delay existing at that time. Notice also that if an attack has already occurred by time t, the individual will not expend on delay-reducing measures at that time, since adhering to the prescribed diet ensures, by assumption, that a second attack will not occur in the future. Substituting Eqs. (10) and (11) into Eq. (9), the Hamiltonian to be maximized is H : (1 − F ){U [c, Y − c − ϕ(D)] − λ(c, S)K} t ¯ + U [1 − F (k)]λ[c(k), S][1 − γ (D(k))] dk + q(1 − F )λ(c, S).
(12)
0
The necessary condition for the evolution of the shadow price over time now becomes q˙ = U (c, h) − λ(c, S)[K − (1 − γ (D))U¯ ] + [δ + λ(c, S)]q,
(13)
whereas the necessary first-order conditions determining c∗ and D∗ are given by Hc = (1 − F ){[Uc (c, h) − Uh (c, h)] − λc (c, S)[K − q − (1 − γ (D))U¯ ]} = 0 (14) HD = (1 − F )[−ϕ (D)Uh (c, h) − λ(c, S)γ (D)U¯ ] = 0.
(15)
A sufficient condition for self-protection, at any given c, is that the marginal benefit from ¯ exceeds the marginal cost of doing so. reducing delay below the public system level D, ¯ this requires that Evaluating condition (15) at D, ¯ U¯ > −ϕ (D)U ¯ h (c, Y − c). λ(c, S)γ (D)
(16)
Evidently, an incentive for self-protection will arise if the marginal expenditure on reduc¯ is sufficiently small. An incentive for self-protection is also more likely ing delay [−ϕ (D)] to arise the greater the risk of suffering an attack [λ(c, S)], the greater the marginal reduction
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¯ or the greater the utility level if surviving an in the risk of dying from an attack [γ (D)], attack (U¯ ). How would high-fat consumption respond to the opportunity of self-protecting against the risk of dying from an attack? To examine this question, suppose, for simplicity, that ¯ ϕ(D) = π(D−D), where π[= −ϕ (D)] denotes the cost of reducing delay by an additional unit of time. Suppose also that π is just high enough so that the individual has no incentive to self-protect (i.e. condition (16) holds as an equality), and consider the effect of a fall in π on D∗ and c∗ . Setting first q˙ = 0 in Eq. (13) and substituting into Eq. (14), yields Hc = Uc (c, h) − Uh (c, h) − λc (c, S) U (c, h) − δ(1 − γ (D))U¯ − λ(c, S)K × K+ = 0. δ + λ(c, S)
(14 )
Totally differentiating Eqs. (14 ) and (15) with respect to D, c, and π , solving for dD∗ /dπ ¯ we obtain and dc∗ /dπ , and evaluating the results at D = D, dD ∗ −Hcc Uh (c, h) ¯ = 2 dπ D=D Hcc HDD − HcD
(17)
HcD Uh (c, h) dc∗ , ¯ = 2 dπ D=D Hcc HDD − HcD
(18)
where the second-order conditions for the maximization of the Hamiltonian require that 2 > 0. While the sign of Eq. (17) is unambiguously positive, imHcc < 0 and Hcc HDD −HcD ¯ the sign of Eq. (18) depends plying that a fall in π at D¯ will induce self-protection (D ∗ ≤ D), ¯ U¯ . 20 on the sign of HcD , which is ambiguous, since HcD = HDc = π(Uch −Uhh )−λc γ (D) Hence, if HcD is negative, self-protection will be accompanied with an increased level of high-fat consumption (therefore, less adherence to the prescribed diet), whereas if HcD is positive, self-protection will be accompanied with a lowered level of high-fat consumption (therefore, more adherence to the prescribed diet). Quite interestingly, and contrary to prior intuition, subscribing to a self-protection scheme must not necessarily give rise to a “moral hazard” effect in the form of stimulating health negligence behavior. That is, dietary adherence and self-protection may be complements in the sense that a fall in price, which induces the latter, enhances the former. It thus follows that public health policy, informed of private preferences, might be able to reduce both the risk of a heart attack and the risk of dying from an attack through subsidizing the price of private emergency services. Rather than trying to change people, public policy could try changing the costs they face.
4. Concluding remarks The present paper has applied a rational expected–utility–maximization model to inquire into the individual’s decision to deviate from a prescribed low-fat diet, despite bearing 20 This may be obtained either through differentiating Eq. (15) with respect to c, or differentiating Eq. (14 ) with respect to D, and substituting Eq. (15).
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an increased risk of suffering a heart attack in the future. The analysis has given rise to a number of interesting conclusions regarding dietary non-adherence and public health policy. First, the risk of undergoing a heart attack assumes a double-agent role in the rational decision model: while acting to deter high-fat consumption, it also enhances non-adherence through discounting future costs at a higher rate, which is positively dependent on the extent of deviation from the recommended diet. In its latter role as a discount factor which increases with the level of non-adherence, the risk of an attack drives the individual to behave less respectfully towards the future. Second, the greater the level of external risk factors, such as smoking, diabetes, obesity, or genetic pre-disposition, the greater may be excess fat consumption, as the shorter the expected time left for the individual to enjoy life. Third, a skillful physician may induce lower consumption of high-fat products through prescribing a stricter than necessary diet. Fourth, contrary to intuition, dietary adherence and self-protection to reduce delay in diagnosis and treatment of an attack can be complements, in the sense that engaging in the latter enhances the former. Public health policy may, thus, help reduce both the risk of an attack and the risk of dying from an attack through providing incentives for high-risk individuals to subscribe to delay-reducing emergency services. The model developed in this paper may be extended in several ways, although at the cost of substantial mathematical complexity. First, rather than assuming that the individual learns his lesson after the first attack, adhering thereafter, to the recommended diet, the possibility of post-attack non-adherence, at the risk of suffering a second attack, may be allowed for. It would be interesting to see whether optimal post-attack adherence is indeed greater than its pre-attack counterpart, or whether it may be rational for the individual to exhibit even a greater disrespect towards his future. Secondly, the possibility of permanent physical and emotional impairments following an attack may be incorporated into the model, rather than restricting its consequences to the extreme cases of death and complete recovery. This could make the model relevant to the case of brain stroke as well, the risk factors for which overlap heavily with those for a heart attack, but which is more likely to result in significant neurological and cognitive impairments. Finally, self-protection measures which reduce not just the risk of dying from an attack, but also the risk of an attack itself, such as cholesterol-lowering drugs (at the cost of unpleasant side effects) or aerobic exercising (at the expense of leisure time), may be incorporated into the model, contributing to a more comprehensive understanding of the non-adherence phenomenon.
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