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:
o highlight that unstable steady states are important for the analysis of addictions,
o explicitly derive long and short run functions for addictive goods,
o discuss the rationality of binges and cold turkey withdrawals, and
o 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
o 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 :
𝑢(𝑡) = 𝑢[𝑦(𝑡), 𝑐(𝑡), 𝑆(𝑡)]

(1)
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:
𝑇
𝑈(0) = ∫ 𝑒 −𝜎𝑡 𝑢[𝑦(𝑡), 𝑐(𝑡), 𝑆(𝑡)]𝑑𝑡
(2)
0
o
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:
𝑆̇(𝑡) = 𝑐(𝑡) − 𝛿𝑆(𝑡) − ℎ[𝐷(𝑡)]
(3)
o Where 𝛿 is an exogenous discount factor on past stock of consumption capital (think
increasing tolerance to alcohol) and ℎ[𝐷(𝑡)] 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
o
o
o
Initial value of assets 𝐴0 ,
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
o
o
𝑇
𝑇
∫ 𝑒 −𝑟𝑡 [𝑦(𝑡) + 𝑝𝑐 (𝑡)𝑐(𝑡) + 𝑝𝑑 (𝑡)𝐷(𝑡)]𝑑𝑡 ≤ 𝐴0 + ∫ 𝑒 −𝑟𝑡 𝑤(𝑆(𝑡))𝑑𝑡
0


(4)
0
Maximizing the given utility function subject to constraints 1 and 2 results in a value function
that represents optimal utility 𝑉(𝐴0 , 𝑆0 , 𝑝) 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, 𝑎(𝑡), and other first order
conditions with respect to the goods in the investment function are
𝑇
𝑇
𝑎(𝑡) = ∫ 𝑒 −(𝜎+𝛿)(𝜏−𝑡) 𝑢𝑠 𝑑𝜏 + 𝜇 ∫ 𝑒 −(𝑟+𝛿)(𝜏−𝑡) 𝑤𝑠 𝑑𝜏
𝑡
(5,6)
𝑡
𝑢𝑦 (𝑡) = 𝜇𝑒 (𝜎−𝑟)𝑡
ℎ𝑑 (𝑡)𝑎(𝑡) = 𝜇𝑝𝑑 (𝑡)𝑒 (𝜎−𝑟)𝑡
𝑢𝑐 (𝑡) = 𝜇𝑝𝑐 𝑒 (𝜎−𝑟)𝑡 − 𝑎(𝑡) = Π𝑐 (𝑡)
𝑑𝑉
where µ = 𝑑𝐴 , which is negative semidefinite, and Π𝑐 (𝑡) is the full price of an addictive good
0
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 𝑎(𝑡) is smaller. The value in this stock
decreases as the stock increases due to the concavity of the value function.
Dynamics:
 Simplifying assumptions:
o Infinite life (𝑇 = ∞)
o Rate of time preference equals rate of interest (𝜎 = 𝑟)
o No endogenous depreciation/appreciation (𝐷(𝑡) = 0)
o Quadratic utility and earnings functions and linear first order conditions
 Utility function quadratic in 𝑦, 𝑆, and 𝐶
 Earnings function is quadratic in 𝑆
o 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
∞
𝑉(𝐴0 , 𝑆0 , 𝑝𝑐 ) = 𝑘 + max ∫ 𝑒 −𝜎𝑡 𝐹[𝑆(𝑡), 𝑐(𝑡)]𝑑𝑡
𝑐,𝑆
(8)
0
where
𝑎𝑐𝑐
𝑎
𝑠𝑠
𝐹(𝑡) = 𝛼𝑐 𝑐(𝑡) + 𝛼𝑠 𝑆(𝑡) +
[𝑐(𝑡)]2 +
[𝑆(𝑡)]2 + 𝛼𝑐𝑠 𝑐(𝑡)𝑆(𝑡) − 𝜇𝑝𝑐 𝑐(𝑡)
2
2
(7)


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.
lim 𝑒 −𝜎𝑡 [𝑆(𝑡)]2 = 0
(9)

The Euler Equation is
𝑡→∞
𝑆̈ − 𝜎𝑆̇ − 𝐵𝑆 =
(𝜎 + 𝛿)𝛼𝑐 + 𝛼𝑠 (𝜎 + 𝛿)𝑝𝑐 𝜇
−
𝛼𝑐𝑐
𝛼𝑐𝑐
(10)
where
𝐵 = 𝛿(𝜎 + 𝛿) +

𝛼𝑠𝑠
𝛼𝑐𝑠
+ (𝜎 + 2𝛿)
𝛼𝑐𝑐
𝛼𝑐𝑐
(11)
The optimal path of capital stock is determined from the initial condition and the smaller of the
two roots, so
𝑆(𝑡) = (𝑆0 − 𝑆 ∗ )𝑒 𝜆1 𝑡 + 𝑆 ∗
(14)
where
𝜆1 =

𝜎 − √𝜎 2 + 4𝐵
2
Characteristics of the optimal path of stock of consumption capital
o The steady state is stable iff 𝐵 = 0, because this forces 𝜆1 = 0
o If the steady state is stable, 𝑆(𝑡) grows (shrinks) over time to 𝑆 ∗ if 𝑆0 < 𝑆 ∗ (𝑆0 > 𝑆 ∗ )
o The path of consumption is
𝑐(𝑡) = (𝛿 + 𝜆1 )𝑆(𝑡) − 𝜆1 𝑆 ∗ ,
(15)
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
(𝜎 + 2𝛿)𝛼𝑐𝑠 ⋛ −𝛼𝑠𝑠 > 0
o

(16)
Therefore adjacent complementarity holds when(𝜎 + 2𝛿)𝛼𝑐𝑠 > −𝛼𝑠𝑠 , that is, the
increase in marginal utility exceeds the increase in full price.
Why is a good addictive to some people and not others?
o 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?)
o This rate of time preference for the individual is key in determining whether adjacent
complementarity exists.

o
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 𝑎𝑐𝑠 > 0
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.
o A future cost is added to the market price if the good is harmful
and subtracted on market price when beneficial.
o 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 :
𝜕 𝜕𝑐(𝑡)
𝜕 𝑑𝑐
𝑑𝑐 𝜕𝑆̇
𝜕 𝑑𝑐
(
𝑆̇) =
+ 𝑆̇
( )
(17)
(
)=
𝜕𝑡 𝜕𝑝𝑐
𝜕𝑝𝑐 𝑑𝑆
𝑑𝑆 𝜕𝑝𝑐
𝜕𝑝𝑐 𝑑𝑆


̇
𝑑𝑐 𝜕𝑆
Because existence of a stable steady state by definition means 𝑆̇=0, only the first term (𝑑𝑆 𝜕𝑝 )is
𝑐
significant.
Because a decline in price implies a rise in consumption of an addictive good, this also implies a
𝜕𝑆̇
rise in the change in consumption stock of the addictive good, so 𝜕𝑝 < 0.
𝑐


The sign of the entire term is opposite that of the change in the change in consumption stock
with respect to price.
o If the sign is positive, the present and past consumption are adjacent complements
o If the sign is negative, the past and present consumption are adjacent substitutes.
o 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.
o

In Figure 2, household consumption with stock of consumption capital to the left of
𝑆 ∗1 but to the right of 𝑆 ∗2 would be to the left of the unstable state when 𝑝𝑐 = 𝑝1 but
to the right of the unstable state when 𝑝𝑐 = 𝑝2 . 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.
o 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
𝑑𝑐 ∗
the good is superior or inferior ( 𝑑𝜇 ≶ 0).



Mini-conclusions
o 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 𝛿 and 𝜎 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.
o








Changes in past prices affect current consumption by changing the stock of
consumption capital.
o 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 𝛿 + 𝜆1
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.
o 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.
o Becker and Murphy account for these life experiences by altering the investment
function to include a vector of each individual’s experiences, called .
o 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.
𝑆̇(𝑡) = 𝑐(𝑡) + 𝑍(𝑡) − 𝛿𝑆(𝑡)
(22)
Temporary events can cause permanent addictions.
o If a person becomes temporarily addicted to a good, they jump from 𝑐1 to 𝑐2 and
eventually travel to 𝑐2′ .
o After the temporary shock is over, the individual’s consumption then returns to the path
𝑝1 𝑝1 and to the consumption point 𝑐1′ .
o 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.
o




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.
o Consumption changes more rapidly when a change in consumption has a larger effect
on future consumption.
o 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.
o If :
 𝑆 < 𝑆̂ , consumption falls from 𝑝3 to 0 over time.
 𝑆 >𝑆̂, consumption rises from 𝑝3 to potentially a stable.
 𝑆 goes from just above 𝑆̂ to just below 𝑆̂, 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).
o 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.