A rational eating model of binges, diets and obesity.

Abstract

This paper addresses the rapid diffusion of obesity and the existence of different individual patterns of food consumption between non-dieters and chronic dieters. I propose a rational eating model where a forward-looking agent optimizes the intertemporal satisfaction from eating, taking into account the cost of changing consumption habits and the negative health consequences of having a non-optimal body weight. Consistent with the evidence, I show that the intertemporal maximization problem leads to a condition of overweightness, and that heterogeneity in the individual relevance of habits in consumption can determine the observed differences in the individual intertemporal patterns of food consumption and body weight. Sufficient conditions for determining when the convergence to the steady state implies oscillations or is monotonic are given. In the former case, the agent optimally alternates diets and binges until the steady state is reached, in the latter a regular intertemporal pattern of food consumption is optimal.

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A rational eating model of binges, diets and obesity
Davide Dragone?
May 26, 2009
Abstract
This paper addresses the rapid di¤usion of obesity and the existence of di¤erent
individual patterns of food consumption between non-dieters and chronic dieters. I
propose a rational eating model where a forward-looking agent optimizes the intertem-
poral satisfaction from eating, taking into account the cost of changing consumption
habits and the negative health consequences of having a non-optimal body weight.
Consistent with the evidence, I show that the intertemporal maximization problem
leads to a condition of overweightness, and that heterogeneity in the individual rele-
vance of habits in consumption can determine the observed di¤erences in the individual
intertemporal patterns of food consumption and body weight. Su¢cient conditions
for determining when the convergence to the steady state implies oscillations or is
monotonic are given. In the former case, the agent optimally alternates diets and
binges until the steady state is reached, in the latter a regular intertemporal pattern
of food consumption is optimal.
JEL classi…cation: I1
Keywords: Body Weight, Food consumption, Habits, Optimal control, Oscillations
1Introduction
Eating behavior exhibits two stylized facts. First, a substantial rise in obesity in the world
population. This trend, that originated in the US, is growing at such an alarming pace that
the World Health Organization begins treating it as a pandemic (Lakdawalla et al., 2005,
Prentice, 2006, Acs and Lyles, 2007, Philipson and Posner, 2008). Second, two typical
patterns of individual food consumption behavior are observed. On the one hand some
individuals are chronic dieters which tend to alternate periods of low caloric intake and
?Dipartimento di Scienze Economiche, Università di Bologna, Strada Maggiore 45, 40125, Bologna,
Italia; tel. +39.051.209.2664; e-mail: davide.dragone@unibo.it. I thank Matteo Cervellati, Luca Lamber-
tini, Arsen Palestini, Luca Savorelli and Paolo Vanin for useful suggestions. The usual disclaimer applies.
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Electronic copy available at: http://ssrn.com/abstract=1436388
periods in which they accumulate body weight, often indulging in binge eating behavior.
On the other hand non-dieters individuals display regular intertemporal patterns of food
consumption and body weight (Fairburn and Wilson, 1993, Polivy and Herman, 1987,
among others).
In this paper I propose a rational dynamic theory of eating behavior where I generalize
the framework proposed in Levy (2002) by allowing for the possibility that individuals
have habits in food consumption. The model delivers both oscillatory and monotonic
intertemporal patterns of individual food consumption and weight, eventually leading to a
condition of overweightness. As a result it allows to rationalize within a unique theoretical
framework the evidence on heterogeneity in individual eating behavior and the evidence
on the observed tendency to obesity.
Eating behavior is a crucial determinant of individual well-being. Consuming food
yields utility and it brings the calories that are necessary for the metabolic activity of the
body. An insu¢cient caloric intake is detrimental to the health of a person as it reduces
the e¢ciency of the physiological activities of the body. When the caloric intake exceeds
the energy requirements, the body stores the excess energy by accumulating body mass
and fat. The consequent increase in body weight can harm individual health if it leads
to a condition of obesity. This condition is well documented to reduce life expectancy by
increasing the probability of developing non-communicable diseases such as heart attacks,
type 2 diabetes, high blood pressure, osteoarthritis and some forms of cancer.
Rational individuals optimally trade-o¤ the utility from eating and the negative health
consequences of having a body weight di¤erent from the physiologically optimal one. The
optimal eating behavior consists of an intertemporal path of food consumption that max-
imises expected lifetime utility. Standard analysis implicitly assumes that the level of food
consumption can be costlessly adjusted at each moment in time, so that any level of food
intake can be chosen. Nonetheless, both empirical evidence and introspection witness the
di¢culties of rapid changes in eating behavior. For example, casual evidence suggests
that for most individuals it is di¢cult to adhere to a diet that sharply alters the amount
of caloric intake. In the model this is rationalized assuming that changing the amount
of food consumption is costly to the agent, and it is interpreted as consumption habit.
When there are consumption habits, sluggish changes in food intake are desirable, but this
can be in contrast with the need of reducing the gap between the actual and the optimal
weight. As a result of this trade-o¤, di¤erent time patterns of eating behavior can emerge.
In the absence of consumption habits, a forward-looking agent maximizes intertemporal
utility by choosing a path of consumption that monotonically drives the agent to the
steady state. With consumption habits, however, following a monotonic path can be too
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costly if it requires too rapid changes in the amount of food consumption. Under these
conditions the optimal consumption path requires a slower convergence to steady state
and it is associated to ‡uctuations above and below the steady state body weight. Due to
sluggish adjustments in consumption, body weight moves toward the steady state weight
but, once this is reached, it is too costly to immediately adjust consumption to the steady
state amount. As a consequence, the steady state is overshot and undershot during the
convergence. The intuition for this result is that, when changing eating behavior is costly,
it is optimal to change it slowly. As a result agents tend to re-adjust their target to
the steady state weight only when they are getting too overweight or too underweight.
For example, an overweight agent who is currently losing weight keeps on restraining
her food intake until she goes underweight and, conversely, an underweight agent who
is gaining weight keeps on getting fat even when the steady state weight is reached. As
getting too thin or too fat is undesirable, a forward looking agent will continue adjusting
consumption until the direction of the path is reversed. The amplitude of these ‡uctuations
decreases overtime until the steady state is reached. This occurs when, in correspondence
of the steady state weight, choosing the steady state amount of food consumption does
not require a large adjustment. Finally if consumption habits are very large, that is if
changing eating behavior is very costly, then oscillations are no longer optimal because
they require too many, and too costly, adjustments. In this case the optimal path requires
a very strict eating behavior leading to a monotonic convergence to the steady state.
The paper delivers several contributions to the literature. The model con…rms the
prediction of Levy (2002) that the steady state equilibrium implies a condition of over-
weightness even in the presence of consumption habits. The analysis however shows that,
in the absence of consumption habits and with general functional forms, the condition of
obesity is actually reached as a steady state with saddle point stability. In such a case
the optimal path of consumption and body weight is monotonic over time.1When indi-
viduals have habits in consumption, in turn, the optimal transition path may also display
an oscillatory convergence to the steady state. This implies that agents …nd it optimal
to alternate binges and diets, and that this oscillatory behavior gets dampened over time
until the steady state is reached. Finally, the analysis provides a novel formalization of
consumption habits which is tractable and intuitive. This approach di¤ers from most of
the literature on habits in consumption, which generally assumes that past consumption
has e¤ect on the marginal utility of current consumption.2The modeling strategy pro-
1The derivation of this result with generic functional forms is presented in the Appendix. For a deriva-
tion using the functional forms adopted by Levy (2002), see Dragone (2009).
2An alternative approach to formalize habits in consumption, originated by the contributions by Boyer
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posed in this paper, in turn, assumes that changes in food consumption are costly. The
emphasis is therefore moved from studying the role of levels of consumption to changes in
consumption. This approach appears well suited to study optimal eating behavior since
people appear to dislike rapid changes in their level of food intake.
The paper is structured as follows. Next section presents the rational eating model
with habits; section 3 shows that the steady state implies a condition of obesity; section
4 studies the asymptotic properties of the steady state and the transition paths; section 5
concludes.
2 A rational eating model with habits
Consider an individual whose utility depends on food consumption c(t) ? 0 according to
the instantaneous utility function U(c(t)), assumed to be strictly increasing and concave.
The existence of habits in consumption is introduced by assuming that changing the level
of food consumption is costly, and that the disutility cost is increasing and convex in
the rate at which this change occurs. Consumption levels and consumption changes are
additively separable, and the instantaneous objective function of the representative agent
is as follows:
V (c(t); _ c(t)) = U (c(t)) ? a_ c(t)2
2
;(1)
where a ? 0 is a constant parameter measuring the marginal disutility of changing the
amount of food intake. When a = 0; we are back to the utility function considered in
Levy (2002), in which habits in consumption play no role.
The probability of survival of the agent depends on her physical weight w(t) > 0 and
on a physiologically optimal weight w?> 0: As in Levy (2002) I assume that deviations
from the optimal weight w?reduce the probability of survival of the agent according to
the function ?> 0; and that this probability is decreasing in its argument
?
of survival be concave in w(t); at least in the relevant domain3. The expected lifetime
?
(w(t) ? w?)2?
(w(t)?w?)2; i.e ?0(?) ? @?(w(t) ? w?)2?
=@
?
(w(t) ? w?)2?
< 0. Let also the probability
(1973, 1983), Pollack (1970, 1976) and Ryder and Heal (1973) and, with reference to addiction, by Becker
and Murphy (1990), Iannacone (1986) and Stigler and Becker (1977), stresses the role of non separabilities
in the levels of consumption at di¤erent points in time, which implies that marginal utility from current
consumption is indeed a¤ected by past levels of consumption.
3Denoting with ?w and ?ww, respectively, the …rst and the second partial derivative of the probability
of survival w.r.t weight, i.e. ?w ? @?(?)=@w and ?ww = @?2(?)=@2w; this amounts to requiring ?ww < 0
for all w(t) in the relevant domain.
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utility of the agent is given by the following expression:
Z1
where ? > 0 is an exogenously given intertemporal discount rate.
0
e??t?
?
(w(t) ? w?)2??
U (c(t)) ? a_ c(t)2
2
?
dt; (2)
To take into account the relation between weight, food consumption and metabolic
needs, the following equation of motion of weight is considered:
_ w(t) = c(t) ? ?w(t)(3)
where ? > 0 represents the rate at which an individual loses weight as a function of the
current weight, i.e. it is a proxy of the individual metabolic needs in terms of individual
weight, which depend on individual characteristics, as well as on individual lifestyle.
Given an initial body weight w(0) = w0and the law of motion (3), the agent must
…nd the intertemporal path of food consumption that maximises (2). The problem has
one control variable, the level of food consumption at time t; and one state variable, body
weight. However, as the objective function contains the rate of change of consumption,
it is convenient to transform the original problem into an equivalent one where there
are two state variables, the body weight and the consumption level, together with the
respective laws of motion, and one new control variable, the rate of change in consumption
(Feicthtinger et al., 1994, Wirl, 1996). Let x(t) denote such a new control variable, so that
x(t) = _ c(t): With this change of variables, the problem of the agent is to select the optimal
rate of change of food consumption that maximizes her discounted lifetime utility, given
the laws of motion of weight (3) and _ c(t) = x(t). Formally, the intertemporal problem
writes as follows:
max
fx(t)g
Z1
0
e??t?
?
(w(t) ? w?)2??
U (c(t)) ? ax(t)2
2
?
dt (4)
subject to
_ w(t)=c(t) ? ?w(t)
x(t)
(5)
_ c(t)= (6)
w(0)=w0;c(0) = c0
(7)
c(t)?0;w(t) > 0: (8)
Notice that this formulation makes clear that, when habits enter the picture, the initial
level of food consumption c(0) becomes relevant in determining the optimal solution,
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while it would not be so when habits do not exist (see the Appendix). The current-value
Hamiltonian corresponding to problem (4)-(8) is:
(w(t) ? w?)2??
where ?(t) and ?(t) are the costate variables representing, respectively, the shadow prices
H(?) = ?
?
U (c(t)) ? ax(t)2
2
?
+ ?(t)[c(t) ? ?w(t)] + ?(t)x(t)(9)
of the dynamics of weight and of consumption.
Assuming that ?ww< (Uc?w)2=(? ? U ? Ucc) < 0 to guarantee that the Hamiltonian
(9) is concave4, the set of necessary and su¢cient conditions for an optimal solution is (to
simplify the notation, the arguments are omitted):
@H
@x
_ w
=? ? ax? = 0 , x =
c ? ?w
x
?
a?
(10)
=(11)
_ c=(12)
_?= ?? ?@H
?? ?@H
@w= (? + ?)? ?
?
U ?ax2
2
?
?w
(13)
_ ?=
@c
= ?? ? ? ? Uc?(14)
Given the in…nite time-horizon, the following transversality conditions are also required
to hold: limt!1e??t?(t)w(t) = limt!1e??t?(t)c(t) = 0:
Given a > 05, the …rst order condition (10) implies that, everything else equal, lower
values of ?(?) determine a higher rate of change in food consumption. This occurs because,
when the gap between current and optimal weight is large, the probability of survival is
low and it is therefore valuable for the individual to quickly adjust consumption in order
to get closer to the optimal weight and to increase the expected lifetime utility. Note also
that changes in the value of a have both a direct and an indirect e¤ect on the rate of
change of consumption x. The direct e¤ect is due to the disutility of habit. Accordingly,
as a increases the rate of change of consumption would slow down to counterbalance the
increased cost of habit. However, a also enters the dynamics of ?; which in turn enters the
dynamics of ?. This implies that there exists an indirect channel through which changes
in the marginal disutility of habits a¤ect x via the shadow cost of habits ?:
4See the Appendix for the details.
5See the Appendix for the case where a = 0.
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Substituting (10) in (12) yields the following system of di¤erential equations:
_ w=c ? ?w
?
a?
(15)
_ c=(16)
_?=(? + ?)? ?
?? ? ? ? Uc?
?
U ?
?2
2a?2
?
?w
(17)
_ ?=(18)
which, together with the initial and the transversality conditions, completely describes the
optimal solution of the agent.
3 Steady state
Without need of making speci…c assumptions on the utility and the survival function, the
qualitative solution of the optimal control problem of the agent can be assessed by studying
the steady state (wss;css;?ss;?ss) that solves with equality the system of di¤erential
equations (15)-(18). First, observe that equation (16) implies that, in steady state, ?ss= 0;
which means, by eq. (10), that the rate of change of steady state consumption is nil. The
level of steady state consumption positively depends on the steady state weight, depending
on the metabolism ? of the agent,
css= ?wss;(19)
so that higher (lower) metabolic needs are associated, ceteris paribus, to higher steady
state consumption. Solving (17) one obtains
?ss=?w(wss)
? + ?
U(css)
Substituting in (18) and equating to zero yields the following implicit relation between
steady state weight and consumption:
?w(wss)U(css) + (? + ?)?(wss)Uc(css) = 0:(20)
As both U(css) and ?(wss) are strictly positive, equation (20) holds if ?w(wss) is negative.
As ?w(wss) = 2(wss?w?)?0(?); this means that the steady state weight corresponds to a
condition of overweightness, wss> w?:
Proposition 1 Given the intertemporal problem (4)-(8), the steady state (css;wss;?ss;?ss)
implies a condition of overweightness for every a.
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The above proposition shows that the optimal solution converges to a condition of
overweightness, irrespective of the individual relevance of habits in consumption, and
therefore it generalizes the proposition in Levy (2002, eq. 14, p. 891) where the attention
is con…ned to the case in which any amount of food consumption can be costlessly chosen,
i.e. a = 0. This is due to the fact that, when consumption and weight are at the steady
state level, habit plays no role because there is no change in consumption. However, as
shown in next section, during the transition habits indeed play a relevant role, as their
marginal impact on the agent critically determine the asymptotic properties of the steady
state and the optimal patterns of body weight and food-consumption over the lifetime.
4Dynamics to the steady state
To determine the asymptotic properties of the steady state, consider the Jacobian associ-
ated to the system of di¤erential equations (15-18):
2
4
2
4
In steady state the Jacobian writes as follows:
J=
6666
@ _ w
@w
@ _ c
@w
@_?
@w
@ _ ?
@w
@ _ w
@c
@ _ c
@c
@_?
@c
@ _ ?
@c
@ _ w
@?
@ _ c
@?
@_?
@?
@ _ ?
@?
@ _ w
@?
@ _ c
@?
@_?
@?
@ _ ?
@?
3
5
7777
=
6666
??
???w
a?2
?2?
??wUc
100
00
1
a?
?ww
2a?2?
?2
a?3
w
?
? U?ww
??wUc
??Ucc
? + ?
?
2a?2?w
??1
3
5
7777
J(wss;css;?ss;?ss) =
2
4
6666
??
0
100
00
1
a?
0?U?ww
??wUc
??wUc
??Ucc
? + ?
?1?
3
5
7777
whose determinant det(J) = ?[(2? + ?)Uc?w+ ? (? + ?)Ucc? + U?ww]=a? is positive
because ?w< 0 by Proposition 1. Using the formula of Dockner (1985), the eigenvalues
of the Jacobian are, for i = 1;2;3;4:
r??
ei=?
2?
2
?2
?K
2?1
2
p
K2? 4det(J); (21)
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where K is the sum of the principal minors of dimension 2 of J :
"
@w @c
Ucc
a
K? det
@ _ w
@w
@_?
@ _ w
@?
@_?
#
+ det
"
@ _ c
@c
@ _ ?
@c
@ _ c
@?
@ _ ?
@?
#
+ 2det
"
@ _ w
@c
@_?
@c
@ _ w
@?
@_?
@?
#
=? ?(? + ?) < 0: (22)
Dockner (1985) shows that, if K is negative and K2=4 ? det(J) > 0; the steady state
is a saddle point with real eigenvalues, two being positive and two being negative, and
therefore the optimal trajectories that converge to the steady state are locally monotonic.
If, instead, K is negative but det(J) > K2=4 > 0; the steady state is a stable focus, the
eigenvalues are imaginary (one pair with positive real part and one with negative real
part) and the optimal trajectories imply dampened oscillations toward the steady state.
In other words, the fact that the determinant of the Jacobian is positive and that K is
negative is su¢cient to show that it is always possible to converge to the steady state
of obesity, the main di¤erence being whether the optimal paths of food consumption are
monotonic or oscillating. Neither explosive behavior leading to starvation or to extreme
binges, nor stable limit cycles where oscillations between diets and binges periodically
alternate over the whole time-horizon can be solutions of the intertemporal maximization
problem under examination.
In the following I show that the conditions that determine whether the steady state
is a saddle point or a stable focus can be ascertained using as a bifurcation parameter
the marginal disutility a of habits in consumption. To see it, observe that the following
expression;
K2
4
? det(J) =[a? (? + ?) + Ucc]2? + 4a[(2? + ?)Uc?w+ U?ww]
4a2?
; (23)
is positive when the numerator is positive. As the numerator is a quadratic function of
a and the coe¢cient of the quadratic term is positive, (23) is positive or equal to zero if
either a ? a1or a ? a2> a1: Let
A= 2(2? + ?)Uc?w+ 2U?ww+ ? (? + ?)?Ucc;
B=[(2? + ?)Uc?w+ U?ww?][(2? + ?)Uc?w+ U?ww+ ? (? + ?)?Ucc];
?2(? + ?)2?;
(24)
C=
then the two bifurcation values are given by the following expression
a1;2= ?A ? 2pB
C
:
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Clearly, A < 0;B > 0 and C > 0 and it can be shown that ?(A + 2pB)=C > 0: As a
consequence both values a1;a2are real, distinct and always positive. The above consid-
erations (together with the case where a = 0, shown in the appendix) can be summarized
in the following proposition:
Proposition 2 Given the intertemporal problem (4)-(8), there exist two bifurcation values
a1;a2, where 0 < a1< a2; such that:
1. if either a 2 [0;a1] or a 2 [a2;1); the steady state is a saddle point and the optimal
transition path of food consumption and body weight is monotonic,
2. if a 2 (a1;a2); the steady state is a stable focus and the optimal transition path of
food consumption and body weight exhibits dampened oscillations.
The nature of the dynamics toward the steady state critically depends on the impor-
tance of habits for the individual. When the marginal disutility of changing the level
of food intake is either low or high, the steady state is a saddle point and the optimal
transition path is monotonic. For intermediate values the steady state is a stable focus
and transition behavior exhibits dampened oscillations. The intuition is the following:
when a is low, monotonicity is optimal because the expected utility gains from a rapid
convergence to the steady state overcome the adjustment costs. As this cost increases,
however, it is optimal to slow down the rate of convergence to reduce the disutility yielded
by the change of habits in consumption. Eventually, this implies that the agent is so slow
in adjusting her eating behavior that, once the steady state weight is reached, she can
prefer to be "conservative" and to choose not to consume the corresponding steady state
level of consumption, but only to get closer to it. This drives the body weight out of
the steady state along the direction that was being followed, i.e. if the person was losing
weight, she keeps on restraining her food consumption, even when she has reached the
optimal weight and, conversely, if the person was gaining weight, she continues intaking
too many calories. As the agent recognizes that getting far from the optimal weight is
detrimental, she will progressively adjust her eating behavior in order to reverse the di-
rection of the path, thereby originating the oscillating behavior that is typically observed
in chronic dieters. Despite the fact that the optimal path does not go straight to the
steady state, it gets closer and closer over time, until it is indeed optimal to choose the
steady state level of consumption when the steady state weight is reached. This result
is consistent with the evidence that eating disorders are more likely to a¤ect adolescents
and young people than older people, which instead exhibit a more regular pattern of food
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consumption and weight. As a further increases, the steady state is again a saddle point.
This occurs because the shadow cost of consumption habits becomes dominant with re-
spect to the disutility induced by habits. In other words, while further increases in the
impact of habit would induce slower and slower consumption changes, also the role played
by the levels of consumption and by body weight on the expected lifetime utility increases
with a; until they overcome the direct costs of habit. When this occurs, oscillations are
no longer optimal because it takes too much time to get to the steady state with respect
to the need of rapidly reducing the gap between current and optimal body weight. As a
result, it is optimal to reach the steady state along monotonic paths of food consumption
and body weight.
5Conclusion
A growing economic literature is addressing the worldwide di¤usion of obesity among both
developed and developing countries, both from a positive and a normative point of view
(Acs and Lyles, 2007, Lakdawalla et al., 2005, Philipson and Posner, 2008, among others).
Most of the economic contributions on the topic stress the role of technological changes and
the importance of price e¤ects in explaining the evidence. This paper takes an alternative
approach and focuses on the intertemporal trade-o¤ between food consumption and the
increasing probability of dying as weight deviates from a given physiologically optimal
level. This perspective allows to deliver a steady state equilibrium that is associated to
a condition of obesity (Levy, 2002), which I show to be always reachable. Additionally,
by assuming that habits determine a utility cost in changing the level of food intake, I
show that heterogeneity in the sensitivity to changes in consumption can account for both
regular patterns of eating behavior, as well as oscillatory behavior between binges and
diets, in a unique theory of rational eating behavior.
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6 Appendix
6.1Concavity of the Hamiltonian function
Consider the Hessian matrix^H(?) associated with the individual objective function:
2
4
Concavity requires Ucc < 0 and U ? Ucc? ? ? ?ww? (Uc?w)2> 0: Clearly, a necessary
condition for the second inequality to hold is ?ww< 0, i.e. the concavity of the survival
function w.r.t w.
^H(v(U(c); _ c)) =
6
Ucc?
Uc?w
0
Uc?w
U?ww
0
0
0
?a
3
5
7
(25)
6.2Saddle point stability when a = 0 (Levy, 2002)
The rational eating model proposed by Levy (2002) can be considered as a special case, for
a = 0; of the model presented in section 2. Levy (2002) concludes that the model admits a
steady state of obesity, and that this steady state is an unstable focus (i.e. the eigenvalues
of the Jacobian matrix are complex with positive real parts). This would imply that the
steady state cannot be reached, and that the optimal paths are characterized by explosive
oscillations of weight and food consumption. However this conclusion is invalid because the
eigenvalues are real and with opposite signs. As a consequence, the steady state is a saddle
point that can always be reached along monotonic optimal paths of food consumption and
weight. This result, that is consistent with Proposition 2, holds in general and does not
depend on speci…c assumptions on the utility function of the agent and the corresponding
survival function.6This is shown below.
6This holds a fortiori with the functional speci…cation used in Levy (2002), where U(c(t)) = c(t)?and
?(?) = ?0e?"(w(t)?w?)2(see Dragone, 2009).
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When a = 0; the intertemporal expected lifetime-utility of the agent is given by the
following expression:
J =
Z1
0
e??tU(c(t))?((w(t) ? w?)2)dt;(26)
The optimal control problem requires maximizing (26) subject to (3). This requires taking
into account the initial weight w(0) = w0and notice that, as any level of consumption can
be costlessly chosen, the initial level of consumption c0is now irrelevant. The current-value
Hamiltonian~H associated to the problem is:
~H = U(c(t))?((w(t) ? w?)2) + ?(t)[c(t) ? ?w(t)] (27)
where ?(t) is the costate variable associated to the dynamic of weight. The set of necessary
and su¢cient conditions is7:
@~H
@c
_ w
=Uc? + ? = 0(28)
=c ? ?w
?? ?@~H
(29)
_ ?=
@w= ?(? + ?) ? U?w
(30)
lim
t!1e??t?(t)w(t) = 0:(31)
Di¤erentiating (28) w.r.t. time and substituting (30) and the value of ? obtained from
condition (28), the following optimal trajectory of food consumption obtains (Levy 2002,
eq. 12, p. 891):
_ c =
1
?Uccf?(? + ?)Uc+ ?w[U ? Uc(c ? ?w)]g (32)
that, together with (29), the initial condition w0 and the transversality condition (31)
completely describes the optimal solution of the problem.
The steady state (wss;css) must simultaneously satisfy the following conditions (see
Levy, 2002, eq. 14, p. 891):
css
=?wss
(33)
?(? + ?)Uc
=??wU > 0 (34)
and therefore the steady state weight corresponds to a condition of obesity.
7Note that concavity is guaranteed under the same conditions stated in the previous subsection.
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To assess the stability of the steady state implicitly de…ned by (33) and (34), consider
"
a21
a22
the Jacobian matrix J =
a11
a12
#
; whose elements are de…ned as follows:
a11
=
@ _ w
@w= ??
@ _ w
@c= 1
@_ c
@w=
a12
=
a21
=
1
Ucc?2fUc[(c ? ?w)?w+ ??] ? U?wg +
+?ww
Ucc?[U ? (c ? ?w)Uc]
1
U2
cc?
a22
=
@w
@c=
?(? + ?)U2
cc? ? [U?w+ (? + ?)Uc?]?
Substituting the steady states conditions (33)-(34) allows to simplify a21and a22as follows:
a21
=
U
(? + ?)Ucc?2
? + ?
?(? + ?)?ww? ? (2? + ?)?2
w
?> 0
a22
=
For a 2 ? 2 Jacobian matrix, the corresponding eigenvalues are given by the following
expression:
e1;2=1
2
Given that tr(J) = ? > 0 and det(J) = ?? (? + ?) ? a21 < 0, the eigenvalues of the
Jacobian are real and with opposite sign, which implies that the steady state is a saddle
h
tr(J) ?
p
tr(J)2? 4det(J)
i
point that can be reached along monotonic optimal paths of food consumption and weight.
15