# Models of Health and Addiction

## Summary and Keywords

One of the implications of the human capital literature of the 1960s was that a great many decisions individuals make that have consequences not just for the point in time when the decision is being made but also for the future can be thought of as involving investments in certain types of capital. In health economics, this led Michael Grossman to propose the concept of health capital, which refers not just to the individual’s illness status at any point in time, but to the more fundamental factors that affect the likelihood that she will be ill at any point in her life and also affect her life expectancy at each age. In Grossman’s model, an individual purchased health-related commodities that act through a health production function to improve her health. These commodities could be medical care, which could be seen as repair expenditures, or factors such as diet and exercise, which could be seen as ongoing additions to her health—the counterparts of adding savings to her financial capital on a regular basis. The individual was assumed to make decisions about her level of consumption of these commodities as part of an intertemporal utility-maximizing process that incorporated, through a budget constraint, the need to make tradeoffs between health-related goods and goods that had no health consequences. Pauline Ippolito showed that the same analytical techniques could be used to consider goods that were bad for health in the long run—bad diet and smoking, for example—still within the context of lifetime utility maximization. This raised the possibility that an individual might rationally take actions that were bad for her health in the long run. The logical extension of considering smoking as bad was adding recognition that smoking and other bad health habits were addictive. The notion of addictive commodities was already present in the literature on consumer behavior, but the consensus in that literature was that it was extremely difficult, if not impossible, to distinguish between a rational addict and a completely myopic consumer of addictive goods. Gary Becker and Kevin Murphy proposed an alternative approach to modeling a forward-looking, utility-maximizing consumer’s consumption of addictive commodities, based on the argument that an individual’s degree of addiction could be modeled as addiction capital, and which could be used to tackle the empirical problems that the consumer expenditure literature had experienced. That model has become the most widely used framework for empirical research by economists into the consumption of addictive goods, and, while the concept of rationality in addiction remains controversial, the Becker-Murphy framework also provides a basis for testing various alternative models of the consumption of addictive commodities, most notably those based on versions of time-inconsistent intertemporal decision making.

Keywords: Grossman model, rational addiction model, health capital, addiction, health investment, health economics

Introduction

The primary theoretical models used in the literature on the economics of health and addiction are both subsets of human capital theory: Michael Grossman’s (1972) model of investment in health capital and Gary Becker’s and Kevin Murphy’s (1988) model of rational addiction. Neither is free of criticism, both from outside the economics profession and from within, but no alternative frameworks can be said to have displaced them. At the very least, both frameworks have become so widely accepted among health economists that they provide the starting point for the development of criticisms and alternative models. The focus in this article, then, is on those models and their place in the economics literature.

Both models, while starting within the broad original frameworks, have been modified over time. Indeed, the original authors of the rational addiction model produced two theoretical versions: one with and one without an explicit addiction capital variable. The analysis that we set out here is necessarily less detailed than might be presented in other settings, so we have modified the structure of the models relative to the original sources for the sake of making the exposition of their key features as clear as possible. We will indicate some of the more significant modifications as we proceed.

We have only touched on the empirical literature on the two models, primarily because it has grown to such a massive degree since the original papers were published. The aim is to try and give a sense of what the human capital approach to health and addiction is about, to highlight some key references for the empirical work that has been done, and to point to possibly fruitful areas for future research.

Health as Capital

The Grossman model of investment in health capital is probably the most commonly cited economic model in the health behaviors literature (Grossman, 1972). Its primary appeal is its capacity to model the demand for healthcare and choices about health-related behaviors, both good and bad, in a dynamic framework; that is, a framework that recognizes that actions taken at one point in time have consequences for future points in time and that what really matters is the cumulative effect of health behaviors, not just behavior in a single period. It starts from a simple utility function, U_{t}(C_{t}, I_{t}, H_{t}) where I is period t consumption of health-related goods, C is period t consumption of non–health-related goods, and H is the individual’s state of health at period t. Elements of the vector I can have both positive and negative effects on health, so the utility function can be written as U_{t}(C_{t}, I_{Gt}, I_{Bt}, H_{t}), where G and B refer to commodities that are Good and Bad for one’s health, respectively. It is common to assume that I_{B} yields positive instantaneous utility, since, otherwise, nobody would actually consume any I_{B}. I_{G} could yield positive or negative utility, on the grounds that things that are good for one are not necessarily enjoyable. Both C and H yield positive marginal utility.^{1}

There are two factors that turn the optimization problem from a static problem into a dynamic one. The first is the fact that Health is a capital good—referred to, in fact, as Health Capital—which accumulates over time and decays gradually. This means that H is not some notion of how healthy one feels today, nor even something like whether one has the flu today, but rather something more fundamental: one’s ability to resist disease, for example.

The equation of motion for H can be written as:

(1)

In (1), δ is the rate at which one’s stock of health capital naturally decays. It is common in theoretical work for this term to be treated as a constant, meaning that, in the absence of any investment in health capital, one’s stock of capital declines at a constant percentage rate of δ per period. This is typically a simplifying assumption—the model can be rewritten with δ as an increasing function of age, for example, with the rate of decay of health capital increasing as one gets older. It can also be written as a decreasing function of H, although in that case it should be written as δ(A_{t}, H_{t-1}) and for realism purposes the increase in δ with age should eventually outweigh the decrease in δ as H increases.

The term F(I_{Gt}, I_{Bt}, H_{t}) is the instantaneous production function for health capital.^{2} Typically it is assumed that the marginal productivity of I_{G} is positive and that of I_{B} is negative, and that F(0, 0, H) = 0 regardless of the value of H. Here, H enters the production function through its impact on the marginal products: the fact that being more healthy today means that one will tend to be more healthy tomorrow is already captured in the (1−δ)H_{t} term. The H term in the health production function can reflect the existence of a maximum, or best possible health state, H_{max}. As one approaches H_{max}, the marginal productivity of I_{G} declines toward zero. Exogenous, or environmental, health factors that could have a negative effect on both the individual’s level of health and the productivity of I_{G} items can also be added to the production function. Since optimal choices with regard to levels of I_{G} items will depend on the marginal productivity of those inputs, levels of productivity-affecting exogenous factors will affect the individual’s decisions about their own health-related behaviors.

The second factor that turns the Grossman model into a dynamic or intertemporal optimization problem is the individual’s awareness that health is a capital good. Given the functional form of equation (1), an increase in I_{Gt} has no immediate payoff: all the benefit is in the future. An increase in I_{G} today, then, increases H tomorrow by an amount that depends on the marginal productivity of I_{G}. Because H is a durable capital good, however, an increase in H_{t+1} by one unit also increases H_{t+2} by (1−δ) units, H_{t+3} by (1−δ)^{2} units and so on. Thus while the return to a unit of I_{G} consumed today does not show up until tomorrow, it also increases H, and therefore utility, relative to what it otherwise would have been, well into the future. In this model a rational individual is defined as an individual who is forward-looking to at least some degree, where the degree to which an individual is forward-looking determines the weight they place on the stream of future health benefits that follow from an additional unit of I_{G} today. If I_{G} yields disutility—that is, if it is something that is good for you but is not pleasant to consume—that disutility will be experienced today, since it is I_{Gt} that enters the period t utility function. Clearly, when the individual is looking to equate the marginal cost and benefit of I_{Gt}, the weight they place on the future benefits will play a crucial role in determining the utility-maximizing position.

The same can be said of consumption of I_{B}. An additional unit of I_{B} consumed today will reduce H, but not until tomorrow, while it yields utility today. The reduction in H tomorrow does not stop there—H will be lower in each future period as a result of today’s consumption of I_{B}. Someone who is not very forward-looking will not place much weight on future health costs and so will tend to consume more I_{B} today. Since that same person will not place much weight on the future health benefits of I_{G}, it is expected that an individual who consumes more I_{B} consumes less I_{G}, and vice versa. In fact, it is expected that she would engage in less forward-looking behavior in general, not just in matters that affect her health, which gives the first broad prediction of the Grossman model—that people who are less forward-looking with regard to investing in their health will be expected to be less forward-looking with regard to other types of investment—investment in education, for example, or investment in financial assets.

The individual’s problem in the Grossman model is to choose values of I_{G}, I_{B}, and C for each value of t, to maximize the present value of her lifetime utility, subject to the equation of motion for health capital, and to the instantaneous budget constraint:

(2)

which, for purposes of the discussion is assumed to be always binding.^{3} A more complete version of the model would replace the instantaneous budget constraint with an equation of motion for financial assets and impose a lifetime no-bankruptcy constraint on it.

Substituting the instantaneous budget constraint into the utility function, the Lagrangean for the dynamic problem becomes (Chow, 1997):

(3)

where β = 1/(1+ρ) is the individual’s subjective discount factor and ρ is her subjective discount rate. Here, λ is the Lagrange Multiplier, which represents the shadow price of health capital. The first order conditions (FOC) for I_{Gt} and I_{Bt} are (using subscript notation on U to denote marginal utilities)

(4)

(5)

(4) and (5) can be written in Marginal Benefit (MB) = Marginal Cost(MC) form, as

(4a)

and

(5a)

In (4a), the FOC for I_{G} in period t, the left-hand side is the MC term, composed of the opportunity cost of I_{G} in terms of C given up plus any disutility associated with I_{G}, U_{Gt} < 0. It is not unusual to assume that U_{G} = 0 and that I_{G} does not enter the utility function directly. The right-hand side is the MB of positive investments in health, composed of three elements: F_{Gt} is the marginal product of I_{G} in increasing H in period t+1, β is the discount factor, which puts the MB term in the same period values as the MC term, and λ_{t+1} is the shadow price of an additional unit of H in period t+1, which takes account of the fact that an increase in H in t+1 will increase H in all future periods, in a pattern that will depend on the rate of depreciation of H, δ. Thus (4a) says that the individual will acquire I_{G} in period t in a manner that equates the current costs with the future benefits. Note that there are two forward-looking elements on the RHS of (4a): the discount factor, β, and the shadow price term, λ. If the individual is not forward-looking at all, the RHS of (4a) will be equal to zero, and the only type of I_{G} commodities she will consume will be those that yield direct current utility: U_{G} > 0. Even then, she will be consuming them only for their enjoyment factor—the pleasure of engaging in certain sports, for example, with no thought to future health benefits.

The same type of interpretation can be given to (5a), but this time it is the MB of a bad—the enjoyment a smoker gets from a cigarette, for example—on the LHS and the RHS is the MC term that combines the current opportunity cost of the C given up in order to buy a unit of B plus the future health consequences of current consumption of I_{B} (remembering that F_{B} is negative). Now if the individual is completely myopic—not forward-looking at all—she will consume I_{B} up to the point where the current MB equals the current marginal opportunity cost, with no thought to the future health consequences of current consumption. Since a forward-looking individual will regard future health consequences of current consumption of I_{B} as a cost term, she will consume less I_{B} than will a myopic individual, and might, if she is at a corner solution, consume none at all.

In addition to the FOC for I_{G} and I_{B}, the Lagrangean expression can be used to find an equation of motion for the shadow price of health:

(6)

Combining equations (4a), (5a), (6), and (1) allows us to eliminate the unobservable term λ and write the problem in terms of three interrelated first-order difference equations, in I_{G}, I_{B}, and H.

For the moment, the focus is on goods that are good for the individual’s stock of health I_{G}, and then attention will be given to consideration of I_{B} type goods. What the individual is doing here is plotting out a trajectory she plans to follow into the future. That is the essence of forward-looking; by focusing on the future the individual is developing a trajectory. If conditions change in the future she may change the trajectory. Even though people tend not to refer to it this way, forward-looking does imply planning a trajectory. In the present case where the focus is just on I_{G}, the trajectory will involve the evolution of both I_{G} and H over time, and indeed the function of I_{G} is really to shape the lifetime trajectory of H. Again, the notion of health capital is key here, because it underscores that how healthy you are in the future depends on health-related activities you undertake in the present. If the individual leaves the task of investing in her stock of health very late, then adjusting her remaining health trajectory can be more difficult and costly.

It can be shown that when dealing with only I_{G} type goods that the individual’s trajectory can be represented by a phase diagram such as Figure 1 with I_{G} on the vertical and H on the horizontal (Laporte, 2014). The lines labeled I_{G}̇ = 0 and Ḣ = 0 are the stationary loci, which determine the evolution of I_{G} and H in different parts of the diagram. The intersection of the two stationary loci is the long run equilibrium of the system but for a finite horizon problem the optimal trajectory for the individual does not take her to that point. This is a basic result of optimal control theory (Ferguson & Lim, 1998, 2003). For an individual who is born healthy—with health stock H_{HIGH}—the optimal trajectory looks like the path Z in Figure 1. The trajectory is drawn on the assumption that $\delta $ is constant; Grossman (1972) also discusses the case where $\delta $ increases as the individual gets older, which will tend to pull up the optimal level of I_{G} later in the trajectory.

Some Theoretical Predictions of the Grossman Model

For an individual who is born healthy there is not much point in investing in I_{G} early on because its marginal productivity will be very low, so H will tend to decline, albeit slowly, from a young age and I_{G} will tend to increase or decrease as necessary to control the rate at which H declines. In effect, the individual knows her initial value of H and knows what she wants H to be at the end of her finite lifetime and uses the trajectory of I_{G} to get from her initial value to her target endpoint for H. This means that her lifetime investment plan will depend very much on her initial level of health. In Figure 1, trajectory W shows the case of an individual with a low initial level of initial health stock, H_{LOW}. A phase diagram can also show the trajectory for an individual who is struck unexpectedly by a major illness that instantly reduces her level of H as in Figure 2.

Figure 2 depicts an individual who is born with a high stock of health. She initially follows trajectory Z but part way through the trajectory she is struck by a significant unanticipated health shock that takes her from health level H_{HIGH} to H_{LOW}. At H_{LOW} the original level of health investment will no longer be optimal so she reoptimizes over the remaining horizon taking H_{LOW} as the initial stock of health for the new plan. She responds to her new lower health state by dramatically increasing her level of I_{G} (point C) and then follows a new optimal trajectory for the remainder of the planning horizon. The amount of the upward jump will depend on how long her remaining horizon is and in particular whether the health shock shortened her life expectancy—the shorter the remaining horizon, the smaller the expected jump.

The Grossman model can also yield a socioeconomic gradient in health. Consider the case of two individuals born with the same high level of H, one of whom has a higher income than the other. Both individuals start from the same initial health stock and both have the same equation of motion for H, but the individual with a higher income faces a lower opportunity cost of investment in health in terms of the value of the consumption utility given up for each additional unit of I_{G} purchased. The fact that the higher income individual will have a higher initial value of I_{G} means that her stock of health will decline more slowly than will that of the lower income individual. As a consequence, the higher income individual will invest via I_{G} from the beginning and the dynamics of H means that the low income individual’s level of H will decline faster than that of the high income individual so that even in the case of two individuals who have the same stock of H at birth, an income-related differential will tend to appear over time.^{4}

Empirical Directions for the Grossman Model

Most empirical implementations of the Grossman model have, by force of data limitations, been static and cross section. Arguably, while they draw on Grossman for inspiration and as a guide to the variables to include—some measure of health status and some measure of medical care utilization, for example—not many authors to date have had the data necessary to allow them to test the structure of the model in detail. With increased availability of individual-level longitudinal health data sets, this limit is being eased significantly.

The Grossman model is, fundamentally, a dynamic model of individual optimization, and this fact should color future empirical implementations. Careful dynamic modeling will be necessary to allow us to identify empirically the structure of the fundamental equations. Consider, for example, the equation of motion for health capital, equation (1):

(1)

In this equation, F(I_{G}, I_{B}, H) is the gross investment function, while net investment—changes in the level of the individual’s stock of health capital—is

(1a)

where ΔH_{t+1} = H_{t+1} − H_{t}.

It was noted earlier that for an individual who is born healthy, the optimal trajectory of H is likely to involve H declining over time, albeit initially slowly, and that the role of the F function is to control the rate of decline of H. This is particularly the case if I_{G} yields disutility on immediate consumption, since in that case, nobody would be consuming it for its own sake. This means that one is likely to observe low values of I_{G} associated with high values of H and vice versa, especially in cross-section data. However, F is a production function and I_{G} does represent beneficial inputs, so the marginal productivity of I_{G} in F will be positive. Thus, increases in I_{G} will increase gross investment in health but the actual time path of H will depend on net investment.

Further, the marginal productivity of I_{Gt} in F will depend on the existing stock of health capital, H_{t}. On the surface, this is nothing unusual—after all, in a standard production problem, the marginal productivity of labor, L, will depend on the stock of physical capital, K, and the stock of K will, in the vast majority of cases, have been accumulated over time as a result of net investment on the part of the firm. The difference here, however, is that whereas in the standard production problem an increase in K will tend to increase the marginal productivity of L, in the case of the gross production function for health capital an increase in H will tend to reduce the marginal productivity of I_{G}. In essence, the healthier you are, the less an additional unit of health investment can add to your level of health. Thus instead of F_{GH} > 0, the hypothesis is that F_{GH} < 0.

In addition to paying attention to the dynamic nature of the model and taking account of that in specifying the hypotheses about such things as the nature of the relation between observed I_{G} and observed H, it is important to pay attention to the various types of I_{G}. The Grossman model is a model of health investment behavior by individuals. Arguably, in its pure form it best fits decisions about diet, exercise, and preventive care. Thus the best bet for estimating it might well be to use data on individual dietary decisions as the dependent variables. Preventive care falls into the same category, but even here it is important to take account of the nature of the commodities involved. Regular mammograms will, for a healthy woman, add nothing to physical H, although they will, if negative for cancer, add to mental H. The same could be said about most check-up type I_{G} goods. On the other hand, regular taking of statins will add to certain individuals’ stocks of H, but only if their cholesterol health was initially below maximal. Much of the existing literature on preventive medical care relates the propensity to use such care to other indicators of degree of forward-looking behavior, so this is an area that, while extending it to longitudinal data has the potential to add significantly to the understanding of the structural coefficients in equations like (1), is already well placed empirically.

Finally, there are curative medical interventions whose marginal productivity becomes positive only in the event of large drops in H. In this case the model yields dynamic predictions about initial curative interventions and the time path of follow-up care based on such factors as the magnitude of the drop in H, the point on the original lifetime trajectory at which the shock occurred, and the expected marginal product of I_{G} in terms not just of improvement in H but also the effect of the improvement in H on individual life expectancy. Thus the decision about whether, what kind, and how much curative and follow-up care to have will, if the model is correct, depend on factors that are already present in the theoretical structure, but estimating the model on individual-level longitudinal data can be expected to sharpen the understanding of the values of those parameters and the interactions among the parameters of the model in the individual’s ultimate decision process quite significantly.

The discussion of the Grossman model has dealt entirely with the nonstochastic version. Obviously, uncertainty plays a major role in the individual’s health investment decisions. This simple statement does not properly represent the range of issues involved in theoretical modeling of uncertainty in a Grossman framework. There are issues pertaining to the type of uncertainty involved—whether uncertainty about one’s health state or about the effectiveness of treatment, for example (Dardanoni, 1988; Dardanoni & Wagstaff, 1990), and about the nature of health shocks. Cropper (1977) considers the case where the individual’s health in any period is given as a result of past health investment decisions, and the probability of her contracting an illness depends on her level of health capital and her stochastic exposure to germs. Liljas (1998) extends the Cropper framework to encompass a finer gradation than simply “ill” or “not ill.” Picone, Uribe, and Wilson (1998), in a simulation framework, and Laporte and Ferguson (2007), in a theoretical stochastic control framework using the Ito calculus for a Poisson process, consider how the probability of being struck by a serious illness (i.e., one that will reduce her stock of health capital) at some point in future years will affect the individual’s lifetime accumulation of health capital, while Laporte and Ferguson (2017), using stochastic control theory for Wiener processes, look at how uncertainty about how the future path of health capital affects health investment decisions.

Nonaddictive and Addictive Consumption

I_{B} type goods can fall under two broad headings—nonaddictive and addictive. For nonaddictive goods, a key citation is Ippolito (1981). In the context of the Grossman utility-maximizing problem, harmful and nonaddictive goods are the I_{B} case already discussed, recognizing that the damage they do to one’s health is permanent and therefore consuming them regularly will have cumulative deleterious effects. They are nonaddictive in the sense that consuming these harmful but nonaddictive goods does not change the consumer’s fundamental preferences. Addictive goods on the other hand, do change the consumer’s preferences.

For addictive goods, the key reference is the Becker and Murphy (1988) model characterizing “rational addiction” (RA). The formal modeling of the B-M model can be nested within the Grossman I_{B} framework set out previously, with some modification to the notation to make the B-M notion of addiction explicit. The economic literature on the consumption of addictive commodities tends to use a different notation from the Grossman literature, so at this point, the notation will be modified to keep it consistent with the B-M literature.

In particular, for consistency with the broader rational addiction literature, I_{Bt} notation is replaced with S_{t} where S is a harmful and addictive good—typically thought of as smoking and measured in terms of number of cigarettes. The concept of a stock of addiction capital, denoted A, can be introduced, which is driven by an equation of motion that expresses the current value of A—the individual’s current stock, or degree, of addiction—as the cumulated effect of past consumption of S. The equation of motion for A can be expressed as

(7)

where g(S_{t}) is a production function representing the degree to which current consumption adds to the stock of addiction capital—the degree to which the individual is addicted to S—and δ is the rate at which addiction capital decays. If the individual quits cold turkey, so that S_{t} = 0 for current and all future t, then the stock of addiction capital will decay at the rate δ per period, and δ can be thought of in terms of how rapidly the cravings die off if a smoker quits altogether, for example. In this version of the model, the instantaneous utility function is usually written as

(8)

Note that the H term has been dropped from the utility function. This is simply a matter of convention: following a suggestion by Becker and Murphy (1988), the production function for H could be modified as

(9)

where accumulated addiction capital enters the production function for health with F_{A} < 0. This notation emphasizes the fact that the health damage done by one more cigarette today is less important than the effect of accumulated past smoking on H. In practice, the RA literature tends to assume that A enters the utility function directly, representing both the health damage of cumulative smoking and the disutility the individual derives from knowing that she is addicted. The RA literature tends to ignore I_{G} type goods as well as nonaddictive I_{B} type goods, and that convention will be followed here.^{5}

Note also that in (8) it is assumed that U_{SA} > 0. This assumption is one of the focal points of the disagreement among health economists and between health economists and addiction specialists from public health disciplines about the acceptability of the B-M model. It states that an increase in A increases the marginal utility which our individual derives from her consumption of S—that is, the more addicted one is, the more pleasure one derives from each additional cigarette. This is a version of the economic concept of adjacent complementarity, and underlies the public health critique that rational addicts are happy addicts. Assuming that A yields disutility (since U_{A}< 0) and that disutility becomes more severe as A increases (U_{AA} < 0), emphasizes that addiction is a bad, not a good, so there are no happy addicts here, but the assumption of adjacent complementarity contradicts the well-established observation from public health that addicts actually get less and less utility from additional cigarettes as their degree of addiction increases—the concept of tolerance. Interestingly, a model of addiction that allows for tolerance is sketched out by Stigler and Becker (1977), although, beyond a discussion by Chaloupka (1991), it has not played much of a role in the RA literature.^{6}

If once again Y_{t} is defined to be the individual’s income in period t, p_{t} to be the price of S relative to C in t, so the price of C is set to 1, and assume that the individual’s instantaneous budget constraint is always binding, so that Y_{t} = C_{t} + p_{t}S_{t}, utility can be written

(10)

and the consumer’s intertemporal optimization problem becomes

(11)

where β = 1/(1+ρ) is again the individual’s subjective discount factor and ρ is their corresponding subjective discount rate.^{7} The term λ is a Lagrange multiplier, in this case the shadow price of addiction capital. Because addiction is a bad, λ will be negative.

The FOCs for the maximization problem are:

(12a)

(12b)

Here (12b) shows the evolution of the shadow price of A as time passes, while (12a) shows the FOC for current consumption of S. The forward-looking aspect of the problem is in the presence of the shadow price of future A on the right-hand side of (12a), which can be written as

(13)

Then the left-hand side of (13) is the MB of another unit of the addictive good consumed now—call it the MB of another cigarette smoked now—while the right-hand side is the MC, which consists of two elements: the opportunity cost of the C that must be given up so that the individual can consume another cigarette, since her budget constraint is assumed to be binding, plus the discounted value placed on the addition to future stock of addiction, which results from smoking another cigarette today. If λ = 0 for all t, the individual places no weight on future addiction stock and the FOC becomes

(14)

which is just the FOC for the optimal consumption of S in a purely static model. Thus a fully myopic consumer of S will behave as though they were solving a static optimization problem, while a forward-looking smoker will have a non-zero value of λ, and hence increase the MC of an additional cigarette today by an amount that reflects the weight placed on future addiction. Clearly, forward-looking smokers could have different values of λ, and place different weights on future consequences of today’s behavior, but all forward-looking smokers will raise the MC of a cigarette today relative to the MC as seen by a fully myopic smoker. While a non-negativity constraint on S has not been built into the expression of the problem, it is clearly the case that an individual who places a lot of weight on future addiction might optimally be at a solution where such a constraint was binding (i.e. might optimally be a non-smoker). If they place enough weight on future addiction capital, they might well be a committed never-smoker, in the sense that no level of p would be small enough to persuade them to smoke. Thus the state variable version of the model can encompass the full range of behavior, from never-consumers of S to fully myopic consumers, depending on the individual’s value of λ.

This model yields a phase diagram as shown in Figure 3. Again, note the nonlinear nature of the individual’s lifetime consumption trajectory for S, the harmful and addictive good. In this case, starting from A = 0 observe that the stock of addiction grows over time because the smoker adjusts her consumption of S over time but does not quit. The model could be modified by putting a non-negativity constraint on S and allowing the optimal level of S to drop to zero before the end of the time horizon, which would be the case of the rational smoker who quits rationally. Once S falls to zero, A will decline at the rate δ. This is consistent with the medical literature, which says that if a smoker quits soon enough their body can undo much of the damage done by their past smoking behavior (Doll, Peto, Boreham, & Sutherland, 2004), although for a warning about how the Doll et al. results should be interpreted, see Peto (2011). The same model could be applied to a heavy drinker who quits and whose degree of liver damage declines starting almost immediately (Cook & Tauchen, 1982).

Is Rational Addiction an Absurd Theory?

The Becker-Murphy model is one of the most controversial models in health economics, engendering disagreement not only among economists but also between health economists and addiction researchers from other disciplines, most notably public health. In particular, it has been argued that the rational addiction (RA) model does not correctly reflect the mechanisms of addiction because it does not allow for tolerance to build up and assumes that all consumers of addictive substances are “happy addicts,” deriving positive benefit from their addiction.

The general interpretation of addiction is that the act of consuming an addictive commodity causes the individual’s demand for it to increase in the future, with no change in its price or any such standard demand shifting factors as income. In other words, an individual who has been consuming an addictive commodity will find herself wanting to consume even more of it in the future. The issue of rationality pertains to whether the individual is aware of this taste-changing factor and whether, in deciding how much of that commodity to consume today, she takes account of its effects on her future desires. An individual who does not take account of the future in making their present-day consumption choices is referred to as naive or myopic, and one who does as rational.

This definition of rationality speaks to the source of the dispute between economists and public health addiction specialists about the term “rational addiction.” Workers in the addictions field typically see, and focus on, people they identify as addicts—people whose consumption of an addictive commodity has become self-destructive. Economists are interested in the entire range of consumption patterns of a commodity with addictive properties, from abstainers through moderate consumers and on to addicts. In the case of alcohol, for example, it is clear that many people can consume in moderation despite its addictive properties while some consume to the point of becoming alcoholics. In modeling the consumption of alcohol at the level of the individual, economists are interested in investigating the difference between these behaviors. Looking at the other most commonly studied addictive commodity, tobacco, it is clear that in North America consumption of cigarettes began to fall almost immediately after the release of the 1964 Surgeon General’s report on smoking and health (US Public Health Service, 1964). This report represented new information, as far as the mass of the public was concerned, about the health risks associated with smoking. The economic model is interested in how the new information caused smokers, and potential smokers, to change their consumption decisions (i.e., the extent to which they were forward-looking and calculating). Econometrically, it is necessary to try and disentangle the effects of new health information from the effects of tax-induced increases in the price of cigarettes, and to factor in the different degrees of addiction that would have been present in the population of smokers. Doing this requires an individual-level theoretical model that is capable of yielding testable predictions about smoking behavior among naive and forward-looking smokers.

Empirical Implementation of the RA Model

The Becker-Murphy RA model yields an estimating equation of the general form:

(15)

where S_{t} is the individual’s current consumption of an addictive good, S_{t−1} and S_{t+1} are past and future consumption levels, P is the price of the commodity, and e reflects the effects of unmeasured lifecycle variables. This is probably the most familiar form of estimating equation, but it must be noted that the form of the estimating equation for the RA model cannot be regarded as carved in stone: Chaloupka (1991) for example, derives two estimation forms, one that explicitly includes the individual’s degree of addiction, A:

(16)

and one that includes both lead and lag consumption and lead and lag prices:

(17)

Interestingly, the lead-lag form is not derived from Becker and Murphy (1988) but from Becker, Grossman, and Murphy (1994) (although that latter paper had been circulating for some time as Becker, Grossman, and Murphy (1990), a National Bureau of Economic Research working paper, before its journal publication). Becker et al. (1994) took as their starting point a period-t utility function of the form U(C_{t}, S_{t}, S_{t−1}, e_{t}), where the term e reflects the effects of unmeasured lifecycle variables on period t utility. It assumed not only that the individual derived positive marginal utility from consumption of S in period t, but also that S displayed adjacent complementarity, in the sense that an increase in S_{t−1} increased the marginal utility of S_{t}. It is this form of utility function that underlies many criticisms of the RA model by noneconomists. The assumption that the marginal utility of S is positive and that the individual is a utility maximizer is part of the basis for the “happy addicts” criticism: the notion that as people increase their consumption of the addictive commodity their utility will increase. Adjacent complementarity, which says that the more of S the individual has consumed in the past, the greater the additional utility they will derive from additional units in the future, appears to violate the well-known proposition of tolerance—the notion that as an individual’s level of addiction increases the satisfaction they get from each unit of the addictive good declines and they must consume more and more of it just to maintain their old level of utility.

One obvious problem with implementing the state-variable version of the model is that A is, for the most part, unobservable. It can be shown, however, that, because A depends on accumulated past S, with some manipulation, it is possible to derive from the solution to the state-variable version of the model, an estimating equation identical to the lead-lag version that was derived from the Becker et al. (1994) version. Thus the two versions of the model are equivalent, and if there are no happy addicts in the state-variable version there are also none in the U(C_{t}, S_{t}, S_{t−1}, e_{t}) version.

While health economists tend to think of the Becker-Murphy model as virtually *sui generis*, it was in fact (as Becker and Murphy made clear in the article) closely tied in to a large body of literature on consumer behavior. The 1970s and 1980s had seen the emergence of a considerable amount of work aimed at making empirical consumer demand studies consistent with economic theory, and recognizing the systems nature of the problem, meaning recognizing that the consumer’s choices had to be tied together by the overall budget constraint. (This literature is reviewed in Barten, 1977, among others.) Habit formation was also a topic of ongoing interest. Theoretical representations were given by Pollak (1970, 1976) who noted that in models of habit formation as they stood at the time, the individual did not take account of the effect of their current purchases on future preferences and future consumption. This clearly meant that the models of the 1970s were only a starting point for the analysis of addiction. As Pollak (1976) noted, the habit formation model was tractable precisely because the individual was completely myopic. Making them forward-looking would have significant implications for welfare analysis—a forward-looking individual could be thought of as maximizing an intertemporal utility function, whereas a fully myopic (or, as some of the other literature referred to it, a naive individual) could not—but would also make their demand behavior much more complex than was the case for the naive, strictly backward looking individual.

Empirically, the most common approach to integrating habits into consumer demand studies at the time was probably based on the Stone-Geary utility function:

(18)

where i indexes the commodities, x_{i} is actual consumption of commodity i, γ_{i} is some measure of necessary consumption of i, and β_{i} is the utility exponent for commodity i. Clearly, if γ_{i} = 0 for all commodities, the Stone-Geary form collapses to a Cobb-Douglas form of utility function. In the Stone-Geary form, the consumer is assumed to get utility only from consumption in excess of some essential level, γ_{i}, which was usually thought of in the first instance as a subsistence level of consumption. The Stone-Geary form of utility function yielded Linear Expenditure System (LES) demand functions of the general form

(19)

where Y is total income, p_{i} is the price of commodity i, and the final sum is overall necessary expenditure. In the LES form, current demand for commodity i was equal to necessary consumption plus some function of discretionary spending—income in excess of that amount that was needed to satisfy necessary consumption. Multiplying through by p_{i} and dividing through by Y gave

(20)

where the dependent variable is now in the form of the share of total income devoted to commodity i. The LES form was convenient for demand system studies because it allowed the imposition of the constraint that the sum of all commodity expenditure shares had to equal 1.

While initially the γ_{i} terms were conceived of as physiological subsistence levels, the concept of necessary expenditure came to be extended to include psychological necessities. This in turn was extended to allow habit formation, by making γ_{i} some function of past consumption of S_{i}, the addictive commodity, so that the utility function for period t became

(21)

(22)

If a commodity was habit forming, an increase in the previous period’s consumption would increase this period’s necessary consumption.

Conceptually, this was easily extended to allow for the replacement of S_{i,t-1} by some measure of cumulative past consumption (see Phlips, 1972), and the state variable approach associated with Houthakker and Taylor (1970) introduced an equation of motion for accumulated habit along the same lines as the equation of motion for addiction capital in equation (4). The simple Stone-Geary form of utility function could then be modified to be

(23)

where A_{t} is accumulated addiction capital. This particular form of utility function shows the unhappiness of the addict explicitly, since utility depends on consumption in excess of the psychological subsistence level, γA_{t}. If the individual tries to settle in at a constant, nonzero level of S, A will continue to build up and (S − γA) will decline as time passes, reducing their utility level despite the fact that there has been no reduction in S. The Stone-Geary form, however, still does not answer the tolerance criticism, since an increase in A, by reducing (S − γA), increases the marginal utility of S. This is what allows addiction to shift the individual’s preferences toward consuming the addictive commodity, but it still suffers from the flaws that it is argued are present in the use of adjacent complementarity to drive consumption.

These formal extensions of consumer theory to incorporate habit formation actually came early in the literature. The issue remained, at this stage in the evolution of the literature, that the individual was maximizing her period-by-period utility, recognizing that past consumption affected current consumption but not explicitly allowing current consumption to affect future consumption and therefore future utility. Lluch (1974) set the habit formation problem up as an optimal control problem, where one of the state variables was a stock of habit capital, built up from past consumption of a habit-forming variable and allowing for depreciation so that the habit would die away radioactively if the consumer cut their consumption to zero in each period. Spinnewyn (1979, 1981) extended Lluch’s analysis, allowing habit formation to affect necessary consumption (or committed consumption, a term often used in the literature that did not adopt a pure Stone-Geary utility function) so that current consumption of a habit-forming good, by adding to the stock of habit capital, would affect necessary consumption of that same good in future periods. Since the effect of accumulated habit was to increase necessary consumption of that commodity in the future, the increased future expenditure could be regarded as a cost of increasing consumption today. Spinnewyn showed that the necessary conditions for the optimal control problem would be written so that the price of current consumption of a unit of a habit-forming good included the future expenditure effect, raising the current price (the full price, from the perspective of a forward-looking consumer) by an amount that depended on the strength of the habit formation effect. A naive consumer of a habit-forming good would not take account of the future effect, so the price the consumer perceived the commodity as having would be lower than the price perceived by a forward-looking consumer. Clearly, this is identical to the interpretation of equation (9). Under rational habit formation, the higher current (full) price would reduce current consumption of the habit-forming good, although the accumulation of habit capital would lead to consumption increasing over time. Thus a forward-looking consumer would tend to allocate a lower share of her current expenditure to the habit-forming good early in her life course consumption plan than would a naive consumer. The analysis was extended by Iannaccone (1986) to focus on the distinction between beneficial and harmful addictions, and to clarify the role of adjacent intertemporal complementarity.

Thus while the theoretical basis for modeling the consumption of addictive goods was well established, by the early 1980s there was an obvious problem with empirical implementation. The pre-Becker-Murphy models offered no clear basis for testing whether consumers of a commodity that was presumed to be addictive were behaving in a rational or a naive way. While it was possible to observe different consumers’ budget allocation and budget shares at any point in time, if rationality manifested itself in lower budget shares for the habit-forming good early in the consumer’s optimal consumption trajectory, there was no obvious way of determining which individuals’ observed budget shares indicated that they were consuming in a forward-looking manner, and which indicated naiveté.

The appeal of the Becker-Murphy framework to applied health economists, then, was not so much in its theoretical advance as in its empirical implications.^{8} The now-familiar lead-lag form of demand function offered a theoretically derived way of distinguishing between forward-looking and naive consumption of addictive commodities. As Becker et al. (1994) put it, “the major difference between [the forward-looking equation] and the myopic equation is that the latter is entirely backward-looking. Current consumption depends only on current price, lagged consumption, the marginal utility of wealth, and current events. Current consumption is independent of both future consumption, C_{t+1} and future events, e_{t+1}” (p. 400).

Is the Rational Addiction Literature in a Rut? Options for Future Directions

While the Becker et al. (1994) version of the RA model has spawned a copious literature, it is not clear what the marginal benefit of recent additions have been. Researchers have been running lead-lag equations on everything from mall shopping to NFL tickets and finding proof of rational addiction (among many other applications of the RA form; see Castiglione & Infante, 2016; Dragone, 2009; Levy, 2009; Olekalns & Bardsley, 1996; Shen & Giles, 2006; Sisto & Zanola, 2005; Spenner, Fenn, & Crooker, 2010). This has undoubtedly added to the skepticism with which the RA model is regarded outside health economics. While the lead-lag form was welcomed as a device for distinguishing empirically between myopic and rational addiction, its track record of success in finding positive coefficients on the lead and lag consumption terms might have become a hindrance to its further development as an empirical strategy.

A number of broad technical issues can be identified. For lack of individual-level data, the early empirical RA literature used aggregate data—state or nationwide. Unfortunately, there has been little attention paid to aggregation issues, in particular to whether a dynamic form that applies to individual consumers can, on a representative agent argument, be applied at aggregated levels. Auld and Grootendorst (2004) show that simply finding positive lead and lag coefficients cannot be taken as evidence of RA, while Gospodinov and Irvine (2005) raise cautions about econometric issues. In addition, it is well known that empirical results that produce positive lead and lag coefficients quite often yield implausible estimates of β, the subjective discount factor. Such estimates of the value of β are so prevalent that Baltagi (2007) has referred to them as the fly in the ointment of the RA model.

One explanation for these odd results might simply be that any population contains a mix of individuals with different strengths of forward-looking behavior. The theoretical RA model yields a lifetime trajectory characterized by saddle-point dynamics, meaning that the lead-lag equation is a second-order difference equation with one stable and one unstable root. These roots can be interpreted in terms of the strength of individuals’ tendencies to be forward- and backward-looking (Chaloupka, 1991). Any given population is likely to contain a wide mix of such individuals, as is clear from the wide range of other behaviors that involve some degree or other of tendency to be forward-looking and seem to be systematically associated with smoking behavior (Evans & Montgomery, 1994; Farrell & Fuchs, 1982; Munasinghe & Sicherman, 2006). Any empirical investigation, then, even when using longitudinal individual-level data, which attempts to fit a single RA equation to the entire sample, can be expected to yield results that depend on the particular mix of individuals in its sample. One approach to investigating this might be by using dynamic quantile regression, as in Laporte, Karimova, and Ferguson (2010). This might also help explain the odd results on discount factors noted by Baltagi (2007).

Another possible avenue for future empirical research would involve bringing the economic RA literature closer to the public health literature in its treatment of tolerance effects and individual responses to those effects. One possible direction for empirical research would involve trying to back out degrees of tolerance from RA equations and match them to the public health view of how individuals evolve tolerance for various addictive substances. This would also permit investigation of the extent to which economic factors—prices and incomes—play into how people respond to the development of tolerances.

Beyond these technical issues, the notion of rational addiction has stimulated a debate on welfare economics. Addiction, in the public health sense, has traditionally been regarded as a bad—something to be corrected. Pure myopic models of addiction clearly lend themselves to arguments in favor of corrective taxes, on the argument that individuals do not take proper account of future consequences of today’s consumption decisions.^{9} Rational addiction, with fully forward-looking individuals, casts doubt on at least some of the basis for imposing cigarette taxes. This has led to a range of criticisms of the RA model, coming from observations of individual health behaviors. Madden (2007) provides a detailed overview of the arguments.

In this regard, there is a growing literature that takes issue with the time consistency assumption in the standard RA model. The quasihyperbolic discounting approach of Gruber and Köszegi (2001) is commonly cited, although they note that they cannot actually distinguish empirically between the two models.^{10} The consumption of addictive commodities (which, as noted earlier, should be differentiated from what the public health literature terms “addiction”) is proving a fruitful area for behavioral welfare economics (see Bernheim, 2016; Bernheim & Rangel, 2007; Chaloupka, 1999; DeCicca, Kenkel, Liu, & Wang, 2016; Jin, Kenkel, Liu, & Wang, 2015). Another promising theoretical approach seems to be the dual-self approach of Fudenberg and Levine (2006) and Fudenberg, Levine, and Maniadis (2014). It is quite clear that, while the RA framework is the dominant framework in the economic literature on the consumption of addictive commodities, its underlying assumptions are still matters of considerable dispute. Although the largest segment of current empirical RA literature may be the estimation of lead-lag consumption equations, the most challenging work may well come from literature that takes the RA structure, both theoretical and econometric, as a starting point for developing criticisms.

Conclusion

A great many of the most important decisions that individuals have to make are forward-looking, in that they require the weighing up of costs and benefits that occur at different points in a finite lifetime of uncertain length. The Grossman and RA models fall neatly into this class of models, which can be treated at a theoretical level using dynamic models and investigated empirically most fully with individual longitudinal data. This does not mean, however, that perfection should be expected on the part of the health investor. The math of compound interest is, at least relatively speaking, a lot easier to understand than the process by which health capital accumulates, in part because how one’s financial capital is accumulating can be observed whereas individuals are not really sure what “health capital” means and really do not have a good sense of how much of it they have until they are hit by a major health shock. Similarly, while the concepts of investment in health capital and investment in education capital derive from Gary Becker’s research program in human capital, it is easier in a broad sense to understand the concept of the returns to investment in education than investment in health.

Nevertheless, individuals are assumed at least to some degree to be forward-looking in their health behaviors, although how they handle that forward-looking problem is still a matter of debate (Gruber & Köszegi, 2001). The addition of uncertainty to formal models of health capital combined with the increased availability of relevant longitudinal data sets should allow for investigation of health investment behaviors not in isolation from but jointly with decisions about financial and educational investment. Combining those categories of forward-looking decision making in a single structure should yield a much better basis for understanding forward-looking behavior, behavior in the face of uncertainty, and adaptation to future uncertainty as joint reflections of individuals’ lifetime planning.

## Further Reading

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## Notes:

(1.)
The original Grossman model did not include what we are terming I_{B} commodities. We have added this term to emphasize that it is possible to analyze, within the same framework, behaviors that are good and bad for one’s health. Keeping the capital accumulation motif in mind, we can think of I_{B}—unhealthy diet and lifestyle—as disinvestment terms. Grossman’s (1972) presentation also did not include I_{G} explicitly in the utility function. Adding it allows the model to be used to consider cases where treatments improve one’s health but are so unpleasant that people are reluctant to use them.

(2.) In the 1972 model, Grossman also included time inputs—in addition to medical commodities, each health input or treatment required a certain amount of time per unit. This required that an additional constraint, representing the individual’s time budget, be incorporated in the model. The production function is often assumed to include education as a productivity-enhancing factor, on the assumption that better educated individuals are better able to process health information and are therefore more efficient at producing health.

(3.)
Strictly speaking, the fact that we treat Y as independent of the individual’s health means that we are dealing with what is commonly termed the consumption version of the Grossman model. If we write Y as Y(H), we are adding the financial investment aspect of the original model. If we combine this with the financial lifetime asset constraint (as opposed to the instantaneous income constraint which we have used here), investment in health can be seen as being an element in the individual’s portfolio allocation decision. In Grossman’s 1972 formulation, which included a time constraint, both sick time (which would be reduced as H increased) and treatment time (which would increase as I_{G} increased) were deductions from total time, being drawn from work and leisure time in a utility maximizing manner. Arguably, the consumption version of the model is the more common one in the literature, although the investment aspect has by no means been dropped from the literature. Empirically, Y(H) may represent very long horizon factors: for example, individuals who take more sick time, and are therefore away from work frequently, may be less likely to receive promotions.

(4.) For details and a graphical depiction see Laporte (2014). In addition to a positively sloped health-income gradient, the literature notes a positive health-education gradient. One ongoing question is whether this arises because better educated individuals are more efficient producers of health, so that we should write Y as Y(H), or whether people who are more forward-looking are more likely to invest in both health and education, even if there is no direct link between them. Since education is presumed to have a positive effect on income, an education-health gradient may still arise if health is a normal good. Econometrically, that case should be easier to control for than either of the other two.

(5.) For an implementation of the version involving both H and A, see Jones, Laporte, Rice, and Zucchelli (2018).

(6.) Becker and Murphy (1988) includes a brief reference to what the public health literature refers to as reinforcement, as well as tolerance.

(7.) We are here assuming time consistent behavior (Strotz, 1955). Whether consumers of addictive goods are in fact time consistent, or whether they display, say, quasi-hyperbolic discounting behavior, is a matter of considerable debate in the literature (Gruber & Köszegi, 2001).

(8.) For a detailed discussion of the difficulty of deriving testable predictions from the rational addiction model, see Caputo (2017).

(9.)
This argument has been applied not only to addictive goods, but to what we have termed I_{B} goods broadly—there is a growing literature on the issue of taxing sugar-sweetened beverages.