Transcript Slide 1

A Theory of Mood Influenced
Investment in Health
Michael Caputo
Amnon Levy
Department of Economics
University of Central Florida
School of Economics
University of Wollongong
Abstract
A mood-utility link is incorporated into a theory of rational investment in
personal health, whereby mood is adversely (positively) affected as
instantaneous utility falls below (rises above) a threshold. The analysis is
conducted within a stochastic dynamic framework, where moods are
experienced along a health-dependent random lifespan and generate returns
that form the individual’s lifetime happiness. The investigation highlights the
effects of the inclination to be moody and the health sensitivity to mood on
investment in one’s personal health.
JEL Classification: D91
Keywords: Utility threshold; Mood; Risk; Exercising; Health; Lifetime
Happiness; Optimal Control
Motivation
•
“Oh what a beautiful morning, Oh what a beautiful day, I got a beautiful feeling,
everything going my way.” (Oscar Hammerstein II and Richard Rodgers, Oklahoma)
•
It is rooted in popular belief that being in a good mood is somehow constructive,
whereas being in a bad mood is destructive. This belief is particularly sensible in
the context of maintaining human, social and political assets.
•
Health is a prime example of a human capital stock affected by mood. Rose (1980)
has found that, measured along a depression-happiness dimension, mood was
ranked by her subjects as more important than the physical and social determinants
of health. A high depression-suicide correlation may reflect how drastic and
prevalent the effect of mood can be.
•
This paper is an attempt to incorporate the effects of mood changes into a theory of
rational decision-making with regard to one’s investment in health.
Scope
•
In a conventional model of lifetime utility from consumption with certain life
expectancy the optimal allocation of one’s time between work and exercising is
found by comparing the marginal income of time spent at work to the product of the
marginal health benefit from time spent on exercising and the marginal contribution
of health to productivity. (Seminal papers: Becker 1965, Grossman 1972)
•
We consider a much broader framework, which implies a much more complex
decision rule. The possible interrelationships between mood, exercising and health
constitute the novel elements of our model.
•
The incorporation of mood into the model is analytically facilitated by assuming an
instantaneous utility threshold below which mood is adversely affected and above
which mood is positively affected.
•
We introduce this influence of the actual-threshold instantaneous utility gap on
mood into the determination of the investment of time in health maintaining and
improving activities.
•
We analyse the effect of mood on the time allocated to these activities with a
stochastic optimal control model in which the sum of the discounted instantaneous
returns on mood—lifetime well-being, or happiness—is assessed with a healthdependent random lifespan.
Wealth
Consumption
Income
Work
Productivity
Exercise
Time
Utility
Health
Survival
Wealth
Mood
Lifetime
Happiness
Central Relationships
•
The four components in the bottom-right corner of the flowchart and the thick solid
arrows that link them to one another and to the rest of the components display the
most novel elements of our analysis.
•
These components suggest that mood directly affects health and is indirectly
affected by health through the effect of health on productivity and thereby on
income, consumption and instantaneous utility.
•
The flowchart also displays the trade off associated with exercising between the
productivity benefits from improved health and the foregone working time.
•
As income and, in turn, consumption and utility are affected by productivity and
time allocated to work, this trade off is further indirectly manifested in mood and
also affected by the effect of mood on health.
•
Improvement in health also increases the probability of survival which, in
conjunction with mood, affects the individual’s expected lifetime happiness.
Mood and its Law of Motion
Mood is a state variable: By definition, mood is a background feeling. Hence, it is
technically correct to treat mood as a state variable.
The intensity of an agent’s mood at time t is denoted by m(t ) with good mood indicated by
m>0 and bad mood by m<0.
There is an individual specific threshold utility level, say u  0 , below (above) which
his/her mood is adversely (positively) affected.
The change in the intensity of one’s mood is assumed, for simplicity, to be symmetric and
proportional to the difference between an agent’s instantaneous utility level and a constant
threshold utility level. Namely,
m(t )   u  c(t )   u 
where   0 indicates the individual’s inclination to be moody; i.e., the intensity by which
the utility gap influences an agent’s mood.
Health Law of Motion
The effect of mood: A good mood improves personal health, whereas a bad mood
deteriorates it. The contribution of mood to the rate of change in the stock of personal
health is
m f (m(t ))
where f ()  C (2) ,  m  0 , f (0)  0 , f (m)  0 for all m  , f (0)  0 , f (m)  0
for all m  0 , and f (m)  0 for all m  0 .
These assumptions indicate that (i) a neutral mood, i.e., m(t )  0 , has no effect on the
rate of change in health, (ii) a mood improvement causes the health stock to increase,
and (iii) there are diminishing effects on the rate of change in health for increasingly
good or bad moods. The parameter  m  0 indicates the sensitivity of the rate of
change in health to mood, or in short, the sensitivity of health to mood.
The effect of exercising: The change in an agent’s health stock is further affected in a
general nonlinear way by the proportion of time that an agent spends exercising,
0  (t )  1, as summarized by
g  (t ) 
where g ()  C (2) , g ( )  0 , and g ( )  0 . Thus an agent’s health stock increases at
a decreasing rate with the time the agent spends exercising per unit time.
The motion of personal health:
h(t )   g  (t )    m f (m(t ))   h(t )
where 0    1 is a fixed rate of natural depreciation.
Work:
Income and Consumption Possibilities
[1  (t )] is the fraction of time at date t allocated to work.
Consequently, income at time t is equal to:
[1  (t )] y  h(t ) 
where y  h(t )  is an agent’s income at time t when the agent spends all of his time working
(i.e., when (t )  0 ).
Assumption: y()  C (2) , y(h)  0 and y(h)  0 . Namely, an agent’s flow of income per
unit time is a strictly increasing and strongly concave function of her health stock.
Assumption: The agent’s income flow per unit time is consumed in its entirety,
c(t )  [1  (t )] y  h(t )  .
The Conflicting Effects of Exercising:
Farsightedness vs. Myopia
•
On the one hand, exercising improves health, which, in turn, increases productivity,
potential earning, prospects of higher future utilities, and better future mood. The
improvement in health also increases the probability of survival and consequently
life expectancy.
•
On the other hand, exercising decreases the time allocated to work (and also
increases the level of fatigue) which, in turn, adversely affects current earning,
consumption, utility, and of course, mood.
•
Myopia is manifested by a paramount consideration for a loss of working time (and,
possibly, energy).
•
We assume that rational people make investment decisions regarding their health—
possibly their most precious asset—in a farsighted manner.
From Experienced Moods to Lifetime Happiness
We postulate that an individual’s lifetime happiness is additively separable in the returns on
moods experienced during any possible lifespan that might extend to time t  0 in the future,
and thus takes the form
t
V (t )   e   m( )d
0
where   0 is an agent’s rate of time preference and  a positive scalar denoting the rate
of return on mood for the individual under consideration.
 transforms mood into well being. For example, an improvement in one’s mood facilitates
self rewarding interactions with family members, friends, colleagues, and business agents.
Uncertain Life Expectancy:
Health Moderated Hazard
The likelihood of dying by time t is given by a cumulative density function satisfying:
0  p(t )  1 , p(0)  0 and limt  p(t )  1.
The hazard function
The probability of an agent dying at time t , given that the agent has survived until that time,
is a strictly decreasing and strongly convex function of an agent’s health:
p (t )
 (h(t ))
1  p(t )
where ()  C (2) , (h)  0 , and (h)  0 (for example, 1/(1  h) ).
By rearrangement, the third state equation of the model is obtained:
p(t )    h(t )  1  p(t )
with initial condition p(0)  0 and terminal condition limt  p(t )  1.
Expected Lifetime Happiness
Expected lifetime-happiness maximizers multiply their accumulated happiness between the
starting point of the planning horizon   0 , to their possible time of death   t , namely
V (t ) , by the probability of dying at time   t , namely, p(t ) .
The sum of all the products of p(t ) and V (t ) associated with any possible time of death
0  t   is an individual’s expected lifetime happiness:
E V (t ) 
def


0


 t  
p(t )V (t )dt   p(t )   e  m( )d  dt .

 0
0
Expected Lifetime Happiness:
The Compact Specification
t
Let a   e   m( ) d and db  p(t )dt , and then integrate by parts to get the following
0
convenient form:
E V (t )  
def


0
 t  

p(t )   e  m( )d  dt
 0

t
 p(t )  e
0



0

 
 m( )d


0
e t 1  p(t )   m(t )dt.

0
e  t  m(t ) p(t )dt
Rational Allocation of Time
The optimal investment in health is the time path (t ) that maximizes the individual’s expected lifetime happiness subject to the laws of motion of mood, health stock and probability of dying, the budget constraint, as well as the initial and terminal conditions on the state
variables. The formal statement of the control problem takes the form:

max
( )
subject to

e t  m(t ) 1  p(t )  dt
0
m(t )   [u([1  (t )] y(h(t ))  u ]
c(t )
h(t )   g  (t )    m f (m(t ))   h(t ) , h(0)  h0
p(t )    h(t )  1  p(t ) , p(0)  0 , limt  p(t )  1
(t )  0t [0, ) .
An optimal solution necessarily satisfies:
H   h g( )  mu [1 ] y(h)  y(h)  0 ,  0 , H   0
h  [   ]h  mu [1 ] y(h)[1 ] y(h)  p(h)[1 p]
m  m  [1 p]  m f (m)h
p  [  (h)]p  m
h   g ( )  m f (m)   h , h(0)  h0
m   u [1  ] y (h)   u  , m(0)  m0
p  (h)[1  p] , p(0)  0 , limt p(t)  1
lim e t H (h, m, p, , h , m ,  p ; )  0
t 
Implications of the Optimality and Slackness Conditions:
Should One Always Exercise?
If h (t )  0 and m (t )  0 , then H  0 as   0 and u(c)  0 , which implies, via the complementary slackness condition, that (t )  0 .
That is, if the current value marginal expected lifetime well-being of health is zero at time t
and the current value marginal expected lifetime well-being of mood is positive at time t ,
then it is optimal not to engage in exercise at that time.
Said differently, if an agent believes that the current stock of health is optimal but would prefer a higher mood state at time t , thereby implying that h (t )  0 and m (t )  0 , then no exercise is optimal in that period.
More generally, if h (t )  0 and m (t )  0 , then no exercise is optimal.
It is important to note, however, that even if h (t )  0 and m (t )  0 hold, it is still possible
that H  0 obtains, in which case no exercise is again optimal. In other words, even if an
agent places a positive marginal value on their health and mood, it is possible for an agent to
rationally choose not to exercise, as an agent’s mood may be sufficiently positive as to generate an increase in health and the benefits it brings even in the absence of exercise.
Furthermore, no exercise is more likely to be optimal when h (t )  0 and m (t )  0 hold the
more prone the agent is to mood swings or displays of moodiness, i.e., the larger is   0 .
Wealth
Consumption
Income
Work
Productivity
Exercise
Time
Utility
Health
Survival
Wealth
Mood
Lifetime
Happiness
Expanded Model:
Incorporating Wealth and More Interactions

V (a , h , m , p , ) def
 max
c (), ()
  ( t  )
e
m(t )[1  p(t )]dt


Subject to:
a(t )  ra(t )  y  h(t ) [1  (t )]  pcc(t ) , a( )  a ,
h(t )  f

(t ), a(t ), m(t );    h(t ) , h( )  h ,
m(t )  u u  c(t ), (t ), a(t ), h(t ), m(t ), p (t )   u    h g  h(t )  , m( )  m ,
p(t )    a(t ), h(t ), m(t ) [1  p(t )] ,
p( )  p , limt  p(t )  1 ,
where   [0, ) is a fixed but arbitrary base period or initial time, V () is the current value
optimal value function, and   ( ,  h , u ,  ,  , pc , r ,  , u) 
def
9
.
The current value Hamiltonian of the expanded optimal control problem is:
F (c, ; )   m[1  p]  Va ()  ra  y(h)[1  ]  pcc   Vh ()[ f ( , a, m;  )   h]
def
 Vm ()  u [u(c, , a, h, m, p)  u]  h g (h)   Vp ()(a, h, m)[1  p].
Under the assumptions that c ()  0 and

(1)
()  (0,1) , the first-order necessary conditions
satisfied by  c ( ),  ()  are given by
Fc (c, ; )   pcVa ()  uVm ()uc (c, , a, h, m, p)  0 ,
(2)
F (c, ; )   y(h)Va ()  Vh () f ( , a, m; )  uVm ()u (c, , a, h, m, p)  0 ,
(3)
while the second-order sufficient condition is that the Hessian matrix of F (c, ; ) with respect to (c, ) , denoted by H F (c, ; ) , is negative definite when evaluated at  c ( ),  ()  ,
i.e.,
def  F
H F c (),  ();    cc
F c


Fc 
uVm ()uc
 uVm ()ucc


,
F  c c ( )  uVm ()u c Vh () f  uVm ()u  c c ( )



( )

( )
is negative definite. Equivalently, all the first-order principle minors of H F  c ( ),  ( );  
are negative and its determinant, denoted by H F  c (),  ();   , is positive. The second-
order sufficient conditions are assumed to hold at  c ( ),  ()  . Thus, by the implicit function theorem,  c ( ),  ()  is the locally unique solution to the first-order conditions..
(4)
Claim 1:
The shadow values of Mood and Wealth are always positive
Proof: Define x  (c, , a, h, m, p) and observe that because ucc (x)  0 and u  0 , the negadef
tive definiteness of H F  c ( ),  ( );   implies that Vm ()  0 .
What’s more, seeing as uc (x)  0 , Vm ()  0 , pc  0 , and u  0 , it follows from the first optimality condition [Eq. (15)] that Va ()  0 . Hence, wealthier individuals are happier in the
sense that their expected lifetime happiness is higher.
In other words, the expected current value marginal lifetime happiness of mood and wealth
are positive, i.e., an increase in an agent’s mood and wealth states in the base period increases
the agent’s expected lifetime happiness.
Claim 2:
The shadow value of Health is not necessarily positive.
Proof: Observe that by the second-order sufficient condition, Vh () f  uVm ()u  0 at
 c (),



() . This condition does not imply that Vh ()  0 , as it is possible for the said
condition to hold even if Vh ()  0 as long as Vh ()   uVm ()u
f , given that u (x)  0 ,
Vm ()  0 , u  0 , and f ( , a, m; )  0 .
In other words, the mathematical structure of our model and the basic assumptions imposed
on the technological and preference functions do not rule out that Vh ()  0 , i.e., an increase
in an agent’s base period health stock may lower the agent’s expected lifetime happiness.
A possible explanation is complacency. That is, an increase in an agent’s base period health
stock diminishes the health-benefit from exercising. Lack of exercising shortens the agent’s
life expectancy and hence engenders loss of years and their associated happiness.
Claim 3: The shadow value of health is positive for people deriving no
positive direct utility from exercising
Proof: A simple rearrangement of the first-order necessary condition (16) yields
y(h)Va ()  uVm ()u (c, , a, h, m, p)
,
(1)
Vh () 
f ( , a, m; )
where it should be recalled that all the terms on the right-hand side of Eq. (18) are positive
save for u (x) , which may be positive, negative, or zero. Clearly, if u (x)  0 , then
Vh ()  0 holds.
That is to say, if the instantaneous preferences of an agent are such that she is either neutral
towards exercise or prefers less of it to more, then her expected lifetime happiness is higher if
she begins in a higher state of health when an optimal plan is made. This is intuitive in that
the agent is exercising but prefers not to, ceteris paribus, hence the only reason for exercising
is that the agent values her health and understands that better initial health will increase her
expected lifetime happiness. We all know people like this—they actively dislike exercise but
know it is good for them and hence do it for health reasons as it is expected to make them
happier.
Claim 4: The shadow value of Health can be negative for a person enjoying
(i.e., deriving positive marginal utility from) Exercising
Proof: An inspection of Eq. (18) shows that a necessary and sufficient condition for
Vh ()  0 is y(h)Va ()  uVm ()u (x) , which implies that u (x)  0 .
Stated plainly, if an agent does not prefer a better base-period stock of health, then the only
reason for engaging in exercise is that she actually prefers more exercise to less.
This type of agent is the “opposite” of that in the preceding slide. A person with such preferences prefers to begin planning with a smaller health stock, but nonetheless engages in exercise because more exercise is preferred to less, meaning that such individuals enjoy exercise
for its own sake independent of any health benefits it provides.
Comparative Dynamics
with Frischian Exercising and Consumption Functions
The signs for all of the comparative dynamics of  c (),  ()  are not clear. As a
result, we continued the comparative dynamics with the Frischian version of
these functions--also known as  -constant demand and supply functions. That is,
we use the Frischian form of the feedback demands  c (),  ()  in order to gain
some qualitative insight into our theory of mood-influenced health.
The
value
of
the
Frischian
consumption
function,
say
ˆ ),
c(
def
  (a, h, m,Va ,Vm , u , pc ) , is the solution of the first necessary condition when Va
and Vm are assumed constant, while the value of the Frischian exercise demand
 (a, h, m,Va ,Vh ,  ) , is the solution of the second necessary
function, say ˆ () ,  def
condition when Va and Vh are assumed constant.
The Effects of Wealth on Consumption and Exercising
cˆ(  )  uVmuca [Vh f  uVmu ]  uVmuc [Vh f a  uVmu a ]

a
H F cˆ(  ), ˆ (  ); 


ˆ ( ) uVmucc [Vh f a  uVmu a ]  u2Vm2ucau c

a
H cˆ(  ), ˆ (  ); 
F


If the marginal utility of consumption is independent of exercise ( uc (x)  0 ),
then an increase in wealth increases the rate of consumption.
If consumption and exercise are complements, i.e., uc (x)  0 , and the net
marginal benefit of exercise is an increasing function of wealth, that is,
Vh () f a ( , a, m; )  uVm ()u a (x)  0 , then an increase in wealth increases the
consumption rate and proportion of time spent exercising.
In other words, wealthier individuals consume and exercise more, and thus
work correspondingly less, under the present suppositions.
If, however, consumption and exercise are substitutes and the net marginal
benefit of exercise is a decreasing function of wealth, then consumption still
rises with wealth but the proportion of time spent exercising falls.
Thus, consumption is normal (in wealth) under either set of conditions, while
exercise is normal under one set and inferior under the other.
(1)
The effect of an increase in an agent’s Health Stock
Recalling Eqs. (25)-(28), if exercise and consumption are complements and exercise and
health are sufficiently strong complements, then good news from one’s physician about one’s
current state of health results in increased consumption, more exercise, and less work. Thus
such individuals do not forego their regular workout and in fact work out more, but they do
partake in a celebratory meal.
If, in addition, exercise is a good and the depreciation rate on health is not too large, then the
good health news improves the flows of health and mood. In view of the aforementioned fact
that Vh ()  0 , the agent is clearly happier because of the good health news.
The effect of an increase in mood state on Consumption and Exercising is qualitatively identical to the effect of an increase in
Wealth
The effect of mood state is given by Eqs. (29)-(32). These calculations, when compared to
Eqs. (21)-(24), demonstrate an unanticipated conclusion, namely, that an increase in an
agent’s mood state is qualitatively identical to an increase in an agent’s wealth, in that one
may replace the word “wealth” with “mood” in the preceding discussion of the comparative
dynamics of wealth, and the qualitative conclusions will emerge unchanged.
Thus, whether one receives good news regarding the performance of one’s portfolio, or simply “wakes up on the good-side-of-the-bed,” an agent responds by changing behavior in a qualitatively identical manner under the given restrictions. Moreover, because Vm ()  0 , the
agent is happier because of the improved mood state.
The effects of the Probability of Dying
The comparative dynamics associated this state variable are given in Eqs. (33)-(36).
If exercise is a complement to both consumption and the probability of expiring, i.e.,
uc (x)  0 and u p (x)  0 , then an increase in the probability of dying increases consumption
and exercise, and health flow either increases more quickly or declines less rapidly.
Mood flow, on the other hand, may decline or improve under the preceding assumptions on
preferences.
Alternatively, if exercise is a substitute for consumption and the probability of expiring, i.e.,
uc (x)  0 and u p (x)  0 , then an increase in the probability of dying still increases consumption, but now exercise and health flow decline, and once again the effect on mood flow
is ambiguous.
Thus consumption rises under either set of assumptions on preferences, but exercise, work,
and health flow behave in qualitatively opposite ways.
The effects of the Value of Mood and Moodiness
Another surprising set of results follows from inspection of Eqs. (45)-(47) and
Eqs. (53)-(55).
An increase in the value that an agent places on her mood state ( Vm ) has qualitatively identical effects on consumption, exercise, work, and health flow, as does
an increase in the sensitivity of the mood flow to the utility gap ( u ), i.e.,
“moodiness”—only mood flow can behave in a qualitatively differ manner.
In particular, if exercise is a good and a complement to consumption, then an
increase in moodiness increases consumption, exercise, and health flow, but decreases work.
On the other hand, if exercise is a bad and a substitute for consumption, then
consumption still rises with moodiness, but exercise and health flow fall and
work rises.
Wealth
Consumption
Income
Work
Productivity
Exercise
Time
Utility
Health
Survival
Wealth
Mood
Lifetime
Happiness
Wealth
Consumption
Income
Work
Productivity
Exercise
Time
Utility
Health
Survival
Wealth
Mood
Lifetime
Happiness
Possible Extensions
• Differentiation between physical health and mental health
with mood mainly affecting mental health and exercising
physical health and with mental health affecting physical
health and vice versa.
• In addition to the duration of exercising, the intensity of
exercising can be taken as a second control variable.