motivation “oh what a beautiful morning, oh what a beautiful day, i got a beautiful feeling,...

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A Theory ofM ood Influenced Investm entin H ealth M ichaelC aputo Am non Levy D epartm entofEconom ics SchoolofEconom ics U niversity ofCentralFlorida U niversity ofW ollongong A bstract 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 dynam ic fram ework, where moods are experienced along a health-dependent random lifespan and generate returns thatform the individual’s lifetim e happiness.The investigation highlights the effects of the inclination to be m oody and the health sensitivity to m ood on investm entin one’spersonalhealth. JEL C lassification:D 91 K eyw ords: Utility threshold; M ood; Risk; Exercising; Health; Lifetim e H appiness;O ptim alControl

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Page 1: 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

A Theory of Mood Influenced Investment in Health

Michael Caputo Amnon Levy Department of Economics School of Economics

University of Central Florida 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

Page 2: 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

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.

Page 3: 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

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 health-dependent random lifespan.

Page 4: 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

Income

Work Utility

Mood

Survival

Productivity

Exercise Health

Time

Consumption

Lifetime Happiness

Wealth

Wealth

Page 5: 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

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.

Page 6: 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

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 0u , 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,

( ) ( )t u c t um

where 0 indicates the individual’s inclination to be moody; i.e., the intensity by which the utility gap influences an agent’s mood.

Page 7: 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

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 (2)( )f C , 0m , (0) 0f , ( ) 0f m for all m , (0) 0f , ( ) 0f m

for all 0m , and ( ) 0f m for all 0m . These assumptions indicate that (i) a neutral mood, i.e., ( ) 0m t , 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 0m 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 ( ) 1t , as summarized by

( )g t

where (2)( )g C , ( ) 0g , and ( ) 0g . 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:

( ) ( ) ( ( )) ( )mh t g t f m t h t

where 0 1 is a fixed rate of natural depreciation.

Page 8: 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
Page 9: 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

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 ( ) 0t ). Assumption: (2)( )y C , ( ) 0y h and ( ) 0y h . 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,

( ) [1 ( )] ( )c t t y h t .

Page 10: 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

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.

Page 11: 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

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 0t in the future,

and thus takes the form

0

( ) ( )t

V t e m d

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.

Page 12: 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

Uncertain Life Expectancy:

Health Moderated Hazard The likelihood of dying by time t is given by a cumulative density function satisfying: 0 ( ) 1p t , (0) 0p and lim ( ) 1t p t .

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:

( )( ( ))

1 ( )

p th t

p t

where (2)( ) C , ( ) 0h , and ( ) 0h (for example, 1/(1 )h ).

By rearrangement, the third state equation of the model is obtained:

( ) ( ) 1 ( )p t h t p t

with initial condition (0) 0p and terminal condition lim ( ) 1t p t .

Page 13: 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

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:

def

0 0 0

( ) ( ) ( ) ( ) ( )t

E V t p t V t dt p t e m d dt

.

Page 14: 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

Expected Lifetime Happiness:

The Compact Specification

Let 0

( )t

a e m d and ( )db p t dt , and then integrate by parts to get the following

convenient form:

def

0 0

0 00

0

( ) ( ) ( )

( ) ( ) ( ) ( )

1 ( ) ( ) .

t

tt

t

E V t p t e m d dt

p t e m d e m t p t dt

e p t m t dt

Page 15: 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

Rational Allocation of Time

The optimal investment in health is the time path ( )t that maximizes the individual’s ex-

pected lifetime happiness subject to the laws of motion of mood, health stock and probabil-

ity 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:

( )

0

max ( ) 1 ( )te m t p t dt

subject to

( ) [ ([1 ( )] ( ( )) ]

( )

m t u t y h t u

c t

( ) ( ) ( ( )) ( )mh t g t f m t h t , 0(0)h h

( ) ( ) 1 ( )p t h t p t , (0) 0p , lim ( ) 1t p t

( ) 0 [0, )t t .

Page 16: 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

An optimal solution necessarily satisfies:

( ) [1 ] ( ) ( ) 0h mH g u y h y h , 0 , 0H

[ ] [1 ] ( ) [1 ] ( ) ( )[1 ]h h m pu y h y h h p

[1 ] ( )m m m hp f m

[ ( )]p ph m

( ) ( )mh g f m h , h(0) h

0

[1 ] ( )m u y h u , m(0) m0

( )[1 ]p h p , p(0) 0 , limt p(t) 1

lim ( , , , , , , ; ) 0t

h m pt

e H h m p

Page 17: 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

Implications of the Optimality and Slackness Conditions: Should One Always Exercise?

If ( ) 0h t and ( ) 0m t , then 0H as 0 and ( ) 0u c , which implies, via the com-

plementary slackness condition, that ( ) 0t . 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 pre-fer a higher mood state at time t , thereby implying that ( ) 0h t and ( ) 0m t , then no ex-

ercise is optimal in that period. More generally, if ( ) 0h t and ( ) 0m t , then no exercise is optimal.

It is important to note, however, that even if ( ) 0h t and ( ) 0m t hold, it is still possible

that 0H 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 gener-ate 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 ( ) 0h t and ( ) 0m t hold the

more prone the agent is to mood swings or displays of moodiness, i.e., the larger is 0 .

Page 18: 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

Income

Work Utility

Mood

Survival

Productivity

Exercise Health

Time

Consumption

Lifetime Happiness

Wealth

Wealth

Page 19: 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

Expanded Model: Incorporating Wealth and More Interactions

( )def

( ), ( )( , , , , ) max ( )[1 ( )]t

cV a h m p e m t p t dt

Subject to:

( ) ( ) ( ) [1 ( )] ( )ca t ra t y h t t p c t , ( )a a ,

( ) ( ), ( ), ( ); ( )h t f t a t m t h t , ( )h h ,

( ) ( ), ( ), ( ), ( ), ( ), ( ) ( )u hm t u c t t a t h t m t p t u g h t , ( )m m ,

( ) ( ), ( ), ( ) [1 ( )]p t a t h t m t p t , ( )p p , lim ( ) 1t p t ,

where [0, ) is a fixed but arbitrary base period or initial time, ( )V is the current value

optimal value function, and def 9( , , , , , , , , )h u cp r u .

Page 20: 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

The current value Hamiltonian of the expanded optimal control problem is:

def( , ; ) [1 ] ( ) ( )[1 ] ( )[ ( , , ; ) ]

( ) [ ( , , , , , ) ] ( ) ( ) ( , , )[1 ].

a c h

m u h p

F c m p V ra y h p c V f a m h

V u c a h m p u g h V a h m p

(1)

Under the assumptions that ( ) 0c and ( ) (0,1) , the first-order necessary conditions

satisfied by ( ), ( )c are given by

( , ; ) ( ) ( ) ( , , , , , ) 0c c a u m cF c p V V u c a h m p , (2)

( , ; ) ( ) ( ) ( ) ( , , ; ) ( ) ( , , , , , ) 0a h u mF c y h V V f a m V u c a h m p , (3)

while the second-order sufficient condition is that the Hessian matrix of ( , ; )F c with re-

spect to ( , )c , denoted by ( , ; )FH c , is negative definite when evaluated at ( ), ( )c ,

i.e.,

def

( ) ( )( ) ( )

( ) ( )( ), ( );

( ) ( ) ( )cc c u m cc u m c

Fc c c cc u m c h u m

F F V u V uH c

F F V u V f V u

, (4)

is negative definite. Equivalently, all the first-order principle minors of ( ), ( );FH c

are negative and its determinant, denoted by ( ), ( );FH c , is positive. The second-

order sufficient conditions are assumed to hold at ( ), ( )c . Thus, by the implicit func-

tion theorem, ( ), ( )c is the locally unique solution to the first-order conditions..

Page 21: 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

Claim 1: The shadow values of Mood and Wealth are always positive

Proof: Define def( , , , , , )c a h m px and observe that because ( ) 0ccu x and 0u , the nega-

tive definiteness of ( ), ( );FH c implies that ( ) 0mV .

What’s more, seeing as ( ) 0cu x , ( ) 0mV , 0cp , and 0u , it follows from the first op-

timality condition [Eq. (15)] that ( ) 0aV . 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.

Page 22: 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

Claim 2: The shadow value of Health is not necessarily positive.

Proof: Observe that by the second-order sufficient condition, ( ) ( ) 0h u mV f V u at

( ), ( )c . This condition does not imply that ( ) 0hV , as it is possible for the said

condition to hold even if ( ) 0hV as long as ( ) ( )h u mV V u f , given that ( ) 0u x ,

( ) 0mV , 0u , and ( , , ; ) 0f a m .

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 ( ) 0hV , 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.

Page 23: 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

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

( ) ( ) ( ) ( , , , , , )

( )( , , ; )

a u mh

y h V V u c a h m pV

f a m

, (1)

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 ( ) 0u x , then

( ) 0hV 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.

Page 24: 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

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

( ) 0hV is ( ) ( ) ( ) ( )a u my h V V u x , which implies that ( ) 0u x .

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 prefe-

rences prefers to begin planning with a smaller health stock, but nonetheless engages in exer-

cise because more exercise is preferred to less, meaning that such individuals enjoy exercise

for its own sake independent of any health benefits it provides.

Page 25: 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

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 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

function, say ˆ ( ) , def (a,h, m,Va ,Vh , ) , is the solution of the second necessary

condition when Va and Vh are assumed constant.

Page 26: 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

The Effects of Wealth on Consumption and Exercising

ˆ [ ] [ ]( )

ˆˆ( ), ( );

u m ca h u m u m c h a u m a

F

V u V f V u V u V f V uc

a H c

(1)

2 2ˆ [ ]( )

ˆˆ( ), ( );

u m cc h a u m a u m ca c

F

V u V f V u V u u

a H c

If the marginal utility of consumption is independent of exercise ( ( ) 0cu x ), then an increase in wealth increases the rate of consumption. If consumption and exercise are complements, i.e., ( ) 0cu x , and the net marginal benefit of exercise is an increasing function of wealth, that is,

( ) ( , , ; ) ( ) ( ) 0h a u m aV f a m V u x , 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.

Page 27: 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

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 ( ) 0hV , the agent is clearly happier because of the good health news.

Page 28: 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

The effect of an increase in mood state on Consumption and Ex-

ercising 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 simp-

ly “wakes up on the good-side-of-the-bed,” an agent responds by changing behavior in a qua-

litatively identical manner under the given restrictions. Moreover, because ( ) 0mV , the

agent is happier because of the improved mood state.

Page 29: 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

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.,

( ) 0cu x and ( ) 0pu x , 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.,

( ) 0cu x and ( ) 0pu x , then an increase in the probability of dying still increases con-

sumption, 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.

Page 30: 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

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 ( mV ) has qualita-tively 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 de-creases 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.

Page 31: 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

Income

Work Utility

Mood

Survival

Productivity

Exercise Health

Time

Consumption

Lifetime Happiness

Wealth

Wealth

Page 32: 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

Income

Work Utility

Mood

Survival

Productivity

Exercise Health

Time

Consumption

Lifetime Happiness

Wealth

Wealth

Page 33: 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

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.