A Theory of Rational Addiction

Gary Becker and Kevin Murphy

Presented by Salama Freed

Introduction:

Are addictions rational?
Becker and Murphy present a framework in which addictions are rational over time.
Model builds on analysis originated by Stigler and Becker (1977) and extended Iannaccone (1984 and 1986).
This publication is the first to:
highlight that unstable steady states are important for the analysis of addictions,
explicitly derive long and short run functions for addictive goods,
discuss the rationality of binges and cold turkey withdrawals, and
show how temporary stressful events can lead to permanent addictions.

The model:

Each individual consumes two goods c and y.
Current utility of the individual depends on a measure of past consumption of c but not of y
The utility of past consumption of c is demonstrated as a stock of “consumption capital” S, which can also be seen as “learning by doing”.
An individual’s utility function is a function of current consumption of c, the stock of consumption capital S, and the consumption of other goods (besides that in which the individual has an addiction), and can be described as :

Consumer utility: Considering a constant rate of time preference (σ), therefore allowing for time separability of the utility function, and length of life T, the individual’s utility function is:

Note that utility is separable in y, c, and S but not y and c alone because the marginal utilities of the consumption of both goods depend on the stock of consumption capital, since it is a function of c.

Constraint 1: The investment function, the change over time of the individual’s stock of consumption capital (which represents the individual’s growing or waning utility from consumption of their vice of choice) is described as:

Where is an exogenous discount factor on past stock of consumption capital (think increasing tolerance to alcohol) and represents the effect of the expenditures on endogenous depreciation or appreciation (for example, Weight Watchers or hypnotism in an attempt to stop a food addiction)

Constraint 2:The budget equation, assuming
Initial value of assets ,
Perfect capital markets,
Earnings at time t, w(S),are a concave function of the stock of consumption capital at time t,
The price of good y is a is numeraire and constant over time,
The rate of interest over time is constant and denoted by r, is

Maximizing the given utility function subject to constraints 1 and 2 results in avalue function that represents optimal utility based on initial assets, initial stock in consumption capital, lifetime earnings, and prices. The value function is concave in initial assets and initial stock of consumption capital.
The shadow price of an additional unit of consumption capital stock, , and other first order conditions with respect to the goods in the investment function are

where , which is negative semidefinite, and is the full price of an addictive good and is the sum of the market price of the good and the value of the anticipated cost or benefit of consumption of the good.

Note the optimal investment in endogenous depreciation (appreciation) of capital stock of a harmful (beneficial) good is largest (smallest) when is smaller. The value in this stock decreases as the stock increases due to the concavity of the value function.

Dynamics:

Simplifying assumptions:
Infinite life (
Rate of time preference equals rate of interest (
No endogenous depreciation/appreciation (
Quadratic utility and earnings functions and linear first order conditions
Utility function quadratic in
Earnings function is quadratic in
Price of addictive good remains constant over time, so the value function is also quadratic.
Using these assumptions allow for simplification of the value function, where now it is only in terms of current consumption and stock in consumption capital.
The optimization problem is now

where

Constraint 1:The investment function (Equation 2 with h=0)
Constraint 2: The transversality condition. At the end of life, the stock in consumption capital is 0. You can’t leave your addiction to your family members. The stock is worthless.

The Euler Equation is

The optimal path of capital stock is determined from the initial condition and the smaller of the two roots, so

where

Characteristics of the optimal path of stock of consumption capital
The steady state is stable iff , because this forces
If the steady state is stable, grows (shrinks) over time to if
The path of consumption is

Is this really representative of addiction?

Adjacent Complementarity and Addiction:

Adjacent complementarity (and distant complementarity) was first introduced by Ryder and Heal in 1973. In short, it is related to the connection between the current level of consumption and the effects of future levels of consumption.
Adjacent complementary implies that someone’s present marginal utility of a good is based on their past consumption of the good.

In this model an individual displays behavior consistent with adjacent complementarity iff they are addicted, i.e., if the individual’s increase in the current consumption of c increases the future consumption of c.

For this model, consumption and capital stock are positively related if

Therefore adjacent complementarity holds when, that is, the increase in marginal utility exceeds the increase in full price.

Why is a good addictive to some people and not others?
The addiction is a result of an interaction between a person and an addictive good. The presence or absence of addiction is best described by the rate of time preference an individual has over the consumption of a particular good (are they willing to sacrifice the future wealth/health for the consumption of that good now?)
This rate of time preference for the individual is key in determining whether adjacent complementarity exists.
Though time preference for the present is not necessary for addiction, individuals with an infinite time preference (no value for the future) are at risk of addiction when
However, there are other factors within the interaction between the person and the addictive good that determines the existence of adjacent complementarity
The rate of depreciation of past consumption
Increases current consumption of the individual to maintain a steady state stock of consumption capital.
Complementarity between past and present consumption
Effect on marginal utility of current consumption of past consumption.
The initial stock of consumption capital
Location of the demand curve
Effect of the stock of consumption capital on earnings
A person who is high or drunk is generally less productive on the job.
Note this is based on the assumption of Stigler and Becker (1977) that assumes that consumption capital has a negative effect on utility and earnings, meaning the full price of the addiction is affected by whether the utility and earnings functions are negative.
A future cost is added to the market price if the good is harmful and subtracted on market price when beneficial.
This suggests that addicts of harmful goods are more present oriented while addicts of beneficial goods are more future oriented.

Permanent Changes in Price:

Suppose the price of an addictive good (either “good” or “bad” addiction) declines in price
This effect of this permanent decline in price over time is represented as :

Because existence of a stable steady state by definition means =0, only the first term (is significant.
Because a decline in price implies a rise in consumption of an addictive good, this also implies a rise in the change in consumption stock of the addictive good, so
The sign of the entire term is opposite that of the change in the change in consumption stock with respect to price.
If the sign is positive, the present and past consumption are adjacent complements
If the sign is negative, the past and present consumption are adjacent substitutes.
If the sign is zero, past and present consumption are independent.
The full effect of a change in price on aggregate consumption of addictive goods includes unstable steady states.
In Figure 2, household consumption with stock of consumption capital to the left of but to the right of would be to the left of the unstable state when but to the right of the unstable state when . This suggests reducing the price of the addictive good will increase the long run demand for the addictive good.

The long run change of a permanent change in price would exceed a short run change until a new stable steady state is reached.
The marginal utility is wealth is negatively related to earnings, so change in the steady state consumption with respect to the marginal utility of wealth determines whether the good is superior or inferior (.
Mini-conclusions
A permanent decline in price of the addictive good would raise the consumption of said good due to concavity in the value function.
The effect of a change in price on addicted goods grows over time when present and past consumption are adjacent complements.
Initially may have small effect on demand, but grows over time to reach a new stable steady state.

If future consumption remains constant, the magnitude of the market price of the addictive good is smaller when and are greater, signifying that the consumption of a harmful addictive good is larger while consumption of a beneficial addictive good is smaller.

Temporary Changes in Price and Life Cycle Events

Past and future prices affect current consumption asymmetrically.
Changes in past prices affect current consumption by changing the stock of consumption capital.
Changes in future prices affect current consumption by changing current prices through the effects on future shocks and future consumption.
Adjacent complementarity is necessary and sufficient for present and future consumption and for present past consumption to be complements.
The signs of both cross-price derivatives depends only on the sign of
A good is addictive iff the consumption of a good at different moments in time are complements and the degree of addiction is stronger when the complementarity is greater.
Anticipated increases in prices lower current consumption and the length of time in which this price change is anticipated affects the magnitude on the effect of consumption. That is, the reduction in past consumption of a good is larger if the price is anticipated to change far into the future.
A temporary change in price is related to the permanent change in price previously discussed.
The complementarity between present and future consumption is larger for more addictive goods, so permanent changes in prices for addictive goods have large effects on their current consumption.
The effect of changes in price can also be linked to the change in consumption due to temporary life shocks, such as divorce, unemployment or other stressors since these stressful events lower the utility of the individual but raise the marginal utility of the addictive good, perhaps by providing relief. So changes in the life cycle have the same effect on consumption as changes in price.
People with the same utility function, wealth and price function still have different degrees of addiction based on their different experiences.
Becker and Murphy account for these life experiences by altering the investment function to include a vector of each individual’s experiences, called .
This also compensates for the idea that those who have never consumed an addictive good have no stock in consumption capital and therefore are at no risk of becoming addicted.

Temporary events can cause permanent addictions.
If a person becomes temporarily addicted to a good, they jump from to and eventually travel to .
After the temporary shock is over, the individual’s consumption then returns to the path and to the consumption point .
However, the individual have accumulated enough consumption capital to still be hooked once the stress has subsided and continues with steady state consumption.

Cold Turkey and Binges

This theory implies that strong addictions can only end with abrupt cessations from the addicted good.
Rational person end their addiction to a good either if events cause his demand to be lowered or if his stock of consumption capital is lowered.
Consumption changes more rapidly when a change in consumption has a larger effect on future consumption.
So rational persons end stronger addictions more quickly than weaker addictions.
If the degree of complementarity and addiction were strong enough, then the original utility function is no longer concave and the relationship between current consumption and stock of consumption capital is discontinuous at a certain point.
If :
, consumption falls from to 0 over time.
, consumption rises from to potentially a stable.
goes from just above to just below , there is a significant drop in the individual’s consumption at the discontinuity point.
This discontinuity is due to the strong complementarity between present and future consumption of the addictive good.
When complementarity is strong, this generates a utility function that is not concave, so there are large swings in consumption with even the smallest stressors in the individual’s environment.
The short run loss in utility is larger for stronger addicts, but rational individuals will sacrifice their short term utility in favor of their long terms gains. Rational addicts will try to find ways to minimize that short run utility loss when they stop cold turkey.
Becker and Murphy also evaluate the rationality of binges by extending their model for the market price of an addictive good. Binges can be generated if the stock of consumption capital is split into two stocks that have different depreciation rates and different degrees of complementarity and substitution.

Summary and Conclusions

Rational addiction implies that an individual seeks to maximize his utility consistently over time.
Goods are potentially addictive if increases in past consumption raise current consumption (adjacent complementarity).
Steady state addiction consumption of addiction levels is unstable with strong adjacent complementarity.
In the long run permanent changes in price causes demands of goods to be more elastic than demand for non-addictive goods.
Temporary changes in the price of an addictive good have smaller effects on current consumption than permanent changes.
Abrupt changes in consumption in the form of cold turkey quitting and binges are consistent with rational behavior with small changes to the model.