University of Wollongong Department of Economics Working Paper Series 2000

Transcription

1 University of Wollongong Department of Eonomis Working Paper Series 000 Rational Non-additive Eating: Cyles, Overweightness, and Underweightness Amnon Levy WP 00-07

2 RATIONAL NON-ADDICTIVE EATING: CYCLES, OVERWEIGHTNESS AND UNDERWEIGHTNESS Amnon Levy University of Wollongong Abstrat Although a deviation from the physiologially optimal weight inreases the probability of dying, the steady state for a lifetime expeted-utility maximiser is a state of overweightness. However, even a small initial deviation from this rationally stationary weight is followed by explosive osillations. These osillations might lead to severe and hroni underweightness in a late stage of life. In the presene of soio-ultural norms of appearane, the rationally stationary weight of fat people is lower than otherwise and the rationally stationary weight of lean people is greater than otherwise. (JEL I1) Corresponding Author: Assoiate Professor Amnon Levy, Department of Eonomis, University of Wollongong, NSW 5, Australia. amnon_levy@uow.edu.au, Tel: , Fax:

3 1. Introdution Cyles in food onsumption (binges and strit diets in partiular) and overweightness and underweightness (obesity and anorexia in partiular) an result from psyhologial, physiologial and environmental problems. They may also reflet attempts to onform to soial and ultural norms of appearane. This paper shows that in a stylised world where there are no psyhologial, physiologial and environmental problems and where no soial and ultural pressures exist the empirially observed overweightness, underweightness and ylial food onsumption an be aused by a rational non-additive eating. In an attempt to explain yles in food onsumption Dokner and Feihtinger (1993) have modified Beker and Murphy s (1988) rational addition model by onsidering two negatively orrelated stoks of onsumption apital: an additive stok and weight. The present analysis presents a different approah: eating is onsidered to be non additive and the possibility of rational ylial food onsumption, as well as overweightness and underweightness, is explained by assuming that utility is derived by onsuming food (taken as a homogeneous aggregate), that for any individual there is a physiologially optimal weight, that the larger the deviation from the physiologially optimal weight the higher the probability of dying, and that a rational person onsiders the risk stemming from beoming overweight, or underweight, and plans his, or her, food-onsumption trajetory so as to maximise his, or her, expeted lifetime utility. A rationally optimal weight trajetory is defined as the weight trajetory assoiated with the food-onsumption path whih maximises the individual s expeted lifetime utility. The rationally optimal weight may not neessarily be equal, or onverging, to the physiologially optimal weight. A positive differene between the rationally optimal weight and the physiologially optimal weight indiates the individual s rationally optimal level of overweightness, whereas a negative weight differential reflets his, or her, rationally optimal level of underweightness. Overweightness and underweightness an be onsidered to be hroni if the rationally optimal weight-trajetory does not 1

4 onverge to the physiologially optimal weight and an be diagnosed as aute if the rationally optimal weight trajetory divergs from the physiologially optimal weight. A manageable, stylised, basi, dynami model of rational, non-additive, eating model is developed. It is shown that when physiologial, psyhologial, environmental and soio-ultural reasons for divergene from a physiologially optimal weight do not exist, the steady state is a state of overweighness. However, it is asymptotially unstable: even a small initial deviation from this stationary weight is followed by explosive osillations and binges whih might lead in a late stage of life to severe and hroni underweightness. The model is extended to the ase where soio-ultural norms of appearane exist. It is demonstrated that in the presene of suh norms the stationary weight of fat people is lower than otherwise and the stationary weight of lean people is greater than otherwise. The paper is strutured as follows. A basi dynami model of rational non-additive eating is introdued in setion. The stationary weight advoated by the model is desribed in setion 3. The possibility of onvergene to, or divergene from, an initial weight to the stationary rationally optimal weight and ylial onsumption is analysed in setion 4. The effet of soio-ulturally preferred weight is analysed in setion 5. Comments about possible extensions of the basi model are made in setion 6.. A basi model of rational non-additive eating It is assumed that the probability of dying ( p) at time t is quadrati in the deviation of weight (W ) from the physiologially optimal weight (W *). Let ρ be the rate of time preferene, T the upper-bound on life expetany, and u a onave instantaneous utility funtion of food onsumption (). Then the deision problem of a rational foodonsumer an be formally presented as T ρτ max p(( W( t) W*) ) e u( ( τ ) dτ dt { } 0 0 t

5 subjet to a weight-motion equation where weight is gained by onsuming food and is lost in proportion to the individual s weight W( t) = ( t) ( W( t)) δ. (1) In this setting δ is a positive salar indiating the marginal, and average, effet of weight on burning alories, and onsequently reduing weight, in performing various physiologial funtions. Integrating by parts, the rational food-onsumer s objetive an be rendered as max e u( ( t)) Φ(( W( t) W*) )] dt { } T 0 ρτ where Φ is equal to 1 minus the umulative density funtion assoiated with p and hene indiates the probability of living beyond t. It is assumed that Φ is diminishing and onave in ( W W*) interior solution exists.. This assumption and the onavity of u ensure that an The orresponding Hamiltonian is ρt H = e u( ( t)) Φ(( W( t) W*) ) + λ( t)[ ( t) δw( t)] () where λ is a o-state variable. The neessary onditions for maximum are: λ ρt = e u( ) Φ (( W W*) ) + λδ W (the adjoint equation) (3) 3

6 ρt e u Φ(( W W*) ) + λ = 0 (the optimality ondition) (4) the weight-motion equation (1) and the transversality ondition λ( T) W( T ) = 0. The neessary onditions and the assoiated singular ontrol imply that the rationally optimal food-onsumption and weight trajetories should satisfy the following noarbitrage rule: u u ΦW W u ΦW + = + Φ u Φ ρ δ. (5).. The impliations of this rule for the rationally optimal stationary levels and trajetories of food-onsumption and weight and the possibility of yles are disussed in the following setions with analytially onvenient expliit speifiations of Φ and u. 3. The steady state: rational overweightness By setting of ċ and Ẇ to be equal to zero equation (5) implies that the stationary levels of the rationally optimal food-onsumption and weight should satisfy u ΦW = + Φ ρ δ. (6) u Let the probability of living beyond t be given by Φ = Φ 0 e µ( W W*) (7) 4

7 where µ is a positive salar and 0 < < 1 Φ 0 is the intrinsi probability of living beyond t (i.e., Φ for a person having the physiologially optimal weight); and let the instantaneous satisfation from eating be given by u = β (8) where 0 < β < 1 is the utility elastiity. Then the stationary levels of the rationally optimal food-onsumption and weight should satisfy ( W W*) = ss ss ( ρ + δ ) β µ. (9) Realling also that the weight-motion equation (1) implies that in steady state ss = δw ss (10) equation (9) an be rendered as a seond-order polynomial of W ss whose only relevant and feasible solution is Wss = 05. W * W * + β( ρ + δ ) / δµ. (11) This result suggests that the rationally optimal stationary weight is greater than the physiologially optimal weight. It also suggests that the rationally optimal stationary level of overweightness rises with the rate of time preferene and the elastiity of utility but delines with the rate of burning alories (δ ) and the rate of deline (µ ) of the probability of living beyond t stemming from an infinitesimal rise in the quadrati deviation from the physiologially optimal weight. 5

8 4. Can rational non-additive eating be ylial? Would it lead to the rationally optimal stationary level of overweightness? To answer these questions let us onstrut the possible trajetories of weight and foodonsumption satisfying equations (1) and (5) in the phase-plane diagram. In realling equations (7) and (8), equation (5) implies that the optimal rate of hange in food onsumption is. ( ρ δ ) = + + µ ( β 1) ( W W *) µδ W + µδw * W β β 1. (1) Hene, the isoline ċ = 0 is given by µ ( β 1) ( ρ + δ ) + ( *) W W µδ W + µδw * W = 0 (13) β and its slope by d dw δβ + δβ 1 β = + β W W ċ = 0 1 * (14) ~ whih is negative for W > W > W * and positive for W < W < W * as displayed in Figure 1, where ~ ~ 1 β W = W * 1 + δβ. (15) 6

9 The diretion of the hange in food onsumption above and below the isoline ċ = 0 is obtained by differentiating equation (1) with respet to W and displayed by the vertial arrows. By virtue of equation (1), the isoline Ẇ = 0 is the lous of all the ombinations of and W for whih = δ W. (16) It is displayed in Figure 1 by a line whose slope is equal to δ. Above this line Ẇ > 0 and below it Ẇ < 0 as displayed by the horizontal arrows. Figure 1 to be inserted here. The rationally optimal stationary level of overweightness is displayed by the intersetion point of the two isolines. The diretions of the horizontal and vertial arrows in the phase-plane diagram suggest that the stationary point an be either a spiral or a entre. The steeper the isoline ċ = 0 and the flatter the isoline Ẇ = 0 (i.e., the smaller δ ) the larger the flutuations around the stationary point. That is, the model suggests that binges and strit diets, and more generally ylial food onsumption and weight, may be rational. As demonstrated in the following, the food onsumption and weight yles are explosive and the stationary level of overweighness is unstable. The linearization of the dynami system onsisting of equations (1) and (1) at the viinity of steady state implies that the x state-transition (or Jaobian) matrix a have the following elements: a 11 = 1 (17a) 7

10 a 1 = δ < 0 (17b) a 1 ρ + δ ss Wss = Wss W δ µ( *) (17) β β 1 β a µ ss 4µδ = + β 1 β ss ( W 0. 5W *) > 0 (17d) ss where the sign of a 1 is likely to be positive for a large value of µ (i.e. large effet of diverging from the physiologially optimal weight on the probability of dying), for a large value of Wss W * (i.e., obesity) and for a low rate of time preferene ρ (i.e., farsighted individual), but negative otherwise. The eigenvalues of a are E1, = 05. ( a11 + a) ± ( a11 + a) 4( a11a a1a1). (18) Positive Negative Realling that the diretions of the horizontal and vertial arrows in the phase-plane diagram imply that the stationary point is either a spiral or a entre, the above disriminant is negative and the eigenvalues are onjugate-omplex pair. Sine their real part a11 + a is positive the stationary point is asymptotially unstable as illustrated in Figure 1 by a diverging spiral. That is, there is no onvergene to the stationary level of overweightness but rather explosive osillations around it. It is illustrated further by the trajetory starting at point A near the steady state that there is a possibility of a hroni loss of weight even below the physiologially optimal level in a late stage of life. The 8

11 greater δ, the steeper the isoline Ẇ = 0 and the greater the likelihood of beoming hronially and fatally underweight. 5. The effet of soial and ultural preferenes on weight Overweightness and underweightness might also stem from onforming to soial and ultural norms and pereptions. For instane, in some oriental ountries overweightness is assoiated with beauty, ontentment, grae and wisdom, whereas in some oidental ountries slenderness is muh more appreiated. If this assoiations between physial attrativeness and weight is valid it is expeted that, eteris paribus, oriental slims and oidental fats are the less fortunate people in their domesti markets of partners. It is shown in the following, however, that despite their failure to onform to the soioultural norms of physial appearane, their weight is loser to the soio-ulturally preferred weight than otherwise. The soio-ulturally preferred weight an be inorporated into the basi model by assuming that the individual suffers from a loss of utility by not onforming to the soioulturally preferred weight (W sp ). In this ase, the rational food-onsumer s objetive an be rendered as max e u( ( t),( W( t) W ) ) Φ(( W( t) W*) )] dt { } T 0 ρτ sp subjet to a weight-motion equation (1). The neessary onditions for maximum and the assoiated singular ontrol equation imply that the rationally optimal food-onsumption and weight trajetories should satisfy the no-arbitrage rule u u W Φ + + uu Φ.. W W u W Φ u + Φ u u W = ρ + δ (19) 9

12 and that in steady state u ΦW + u Φ u u W = ρ + δ. (0) In order to ompare the stationary weight in this ase (W ss ' ) to the stationary weight ase where no soio-ultural pressure exists (W ss ), the probability of living beyond t is speified as depited earlier by equation (7) and the instantaneous utility from eating desribed by equation (8) is deflated by not onforming to the soio-ulturally preferred weight as follows: u( t) = β ( W( t) ) W sp. (1) These speifiations imply that the stationary weight when there is a soio-ulturally preferred weight should satisfy W '[ µ ( W ' W*) + ( W ' W ) ] =. β( ρ + δ ) / δ ss ss ss sp 1 05 () whereas the stationary weight when there is no soio-ultutrally preferred weight should satisfy W [ µ ( W W*)] = 0. 5 β( ρ + δ ) / δ. (3) ss ss If Wss '> W sp, whih is the likely harateristi of oidental people who are less suessful in the mathing game, then Wss '< W ss. That is, when there exists a soio- 10

13 ultural norm of appearane their stationary weight is lower than otherwise. If, however,wss '< W in the mathing game, then Wss sp, whih is the likely ase of oriental people who are less suessful '> W of appearane their stationary weight is greater than otherwise. ss. That is, when there exists a soio-ultural norm 6. Conlusion The paper presented a manageable stylised basi model of non-additive eating that an explain overweightness, underweightnes and ylial food onsumption by assuming that the satisfation from eating is ounterbalaned by inreasing probability of dying as weight deviates from the physiologially optimal level. It was found that when physiologial, psyhologial, environmental and soio-ultural reasons for divergene from a physiologially optimal weight do not exist, the steady state for a lifetime expeted-utility maximiser is a state of overweighness. The rationally optimal stationary level of overweightness was shown to be rising with the individual s rate of time preferene and elastiity of utility but delining with his, or her, rate of alories burning and rate of deline of the probability of ontinuing living aused by an infinitesimal rise in the quadrati deviation from the physiologially optimal weight. It was shown, however, that even a small initial deviation from this rationally optimal stationary weight is followed by explosive osillations, i.e., binges and strit diets, and possibly a severe and hroni underweightness in a late stage of life. The inorporation of soio-ultural norms into the basi model revealed that when there exists a soio-ultural norm of appearane the stationary weight of fat people is lower than otherwise and the stationary weight of lean people is greater than otherwise. Some other possible effets of deviation from the physiologially optimal weight and budget onsiderations were not inorporated into the model for sake of simpliity. In a broader framework, instantaneous utility may be presented as derived from eating and devoting a fration (l) of an instane to leisure ativities (the rest, 1 l, is devoted to inome-generating ativities), and a deviation from the physiologially optimal weight 11

14 might also lead to a loss of satisfation from leisure ativities and a loss of effiieny in generating inome. Intuitively, it is expeted that the inorporation of these elements would redue the differene between the rationally optimal stationary weight and the physiologially optimal weight, dampen the osillations of the food-weight joint trajetory, and inrease the likelihood of onvergene to the stationary weight. An extension of the basi model along these lines is desribed in the Appendix. However, the omplexity of the extended model rendered the assessment of this intuitive assertion and the properties of the optimal trajetories of food onsumption, leisure and weight impossible. Referenes Beker, Gary. S., and Murphy, Kevin M., A Theory of Rational Addition, Journal of Politial Eonomy, August 1988, 96, Dokner, Engelbert. J., and Feihtinger, Gustav, Cylial Consumption Patterns and Rational Addition, Amerian Eonomi Review, Marh 1993, 83,

15 APPENDIX: An Extension of the Basi Model An intensive effort has been made in extending the basi model. By inorporating the elements indiated in the onluding setion into the basi model, the modified deision problem of a rational food-onsumer an be portrayed as follows: T max e ρt u ( ( t ),[ ( W ( t ) W *) ] l( t )) (( W ( t ) W 1 γ Φ *) ) dt {, l} 0 where weight is gained by onsuming food and is lost by spending energy on work and leisure ativities and proportionally to its urrent level W( t) = ( t) [ δ l + δ ( l )] W( t) (A1) 1 1 and where ( t) = [ 1 θ( W( t) W*) ] f ( 1 l( t)). Inome (A) In this framework, δ 1 and δ are positive salars indiating the marginal effet of weight on burning alories, and subsequently weight, in leisure ativities and work, respetively; γ and θ are positive salars indiating the marginal effets of being overweight or underweight on satisfation from leisure ativities and on inome generation, respetively; and f is a onave earning funtion ( f '> 0 and f "< 0 ) for a person having physiologially optimal weight. 13

16 Finally, the degree of omplexity was largely inreased when equation (7) was modified as follows Φ = Φ 0 e µ ( W ( t ) W *) T t (A3) to inorporate the possible effet of age and the moderating influene of the distane from the upper-bound on life expetany on the adverse effet of a deviation from the physiologial weight on the probability of living beyond t. 14

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