2.1. The economy

We build a continuous time dynamic general equilibrium model for a closed economy in which the capital accumulation technology exhibits decreasing returns. There is a large number of firms and households, the respective number of which is normalized to unity. Households derive utility from two types of consumption: food consumption that does not require time, and exercise-related consumption that requires time and is affected by peer effects. In our base model, households maximize utility subject to an intertemporal budget constraint, and body weight is the result of optimal food and exercise choices. We discuss the case when the effect of net calorie intake on body weight gain is endogenized in Section 3.3. In what follows, the time index t is suppressed, unless needed for clarity.

2.1.1. Time-consuming consumption

First, the representative individual distinguishes between two types of goods, those that do not require time, $C$ (food consumption) and those that require time, $X$ (exercise). Second, we introduce endogenous labor in the model. Individuals are endowed with one unit of time, used for endogenous labor $N$ , endogenous exercise $X$ , and exogenous sedentary leisure $\bar{S}$ (such as sleeping or watching television). As a consequence, we consider the time constraint:

(1) \begin{equation} 1=N+X+\bar{S}\,. \end{equation}

To simplify the notation, we write that the amount of time that is not spent on sedentary leisure $\bar{L}=1-\bar{S}$ , such that:

(2) \begin{equation} \bar{L}=N+X \end{equation}

As a consequence, an individual’s flow budget constraint accounts for both work time and consumption expenditure related to exercise:

(3) \begin{equation} \dot{K}=rK+w\left (\bar{L}-X\right )-p_{C}C-p_{X}X\,,\,\,\,p_{X},\,p_{C}\gt 0\,, \end{equation}

where $w$ is the wage rate, $r$ denotes the interest rate, and $p_{C}$ and $p_{X}$ are the respective exogenous prices of $C$ and $X$ .

2.1.2. Peer influence

Individuals are influenced by peers when choosing to exercise. We introduce a reference level of exercise, $\bar{X}$ , to take into account such peer influence. In our model, the impact of the reference level of exercise is captured by effective exercise $\hat{X}$ , which differs from absolute exercise $X$ according to the standard subtractive specification [Ljungqvist and Uhlig (Reference Ljungqvist and Uhlig2000)]:Footnote ³

(4) \begin{equation} \hat{X}=X-\varepsilon \left (k\right )\bar{X}\,,\,\,\,\,\,0\leq \varepsilon \left (k\right )\leq 1, \end{equation}

where $k\equiv K/N$ denotes aggregate wealth per unit of labor (roughly, capital per worker), which is exogenous for an individual. The reference level $\bar{X}\equiv X/1$ is given by average exercise expenses (aggregate exercise expenses, with the population size equaling unity). It is endogenously determined in our model, but it is exogenous from an individual’s point of view (as indicated by the upper bar). The degree (strength) of peer influence is captured by function $\varepsilon \left (k\right )$ . When $\varepsilon \left (k\right )=0$ , individuals are not influenced by peers, and effective exercise equals absolute exercise. When $\varepsilon \left (k\right )$ is high, effective exercise is low, and the marginal utility of $X$ is high. The formulation of peer effects is similar to Mathieu-Bolh and Wendner (Reference Mathieu-Bolh and Wendner2020) in the sense that it includes an exogenous and an endogenous element. We use the following standard functional form to describe the degree of peer influence:

(5) \begin{equation} \varepsilon (k)=1-e^{-\kappa k},\quad \kappa \gt 0. \end{equation}

The static element $\kappa$ yields the property called the static peer effect. In our model, it means that, given the stock of capital $k$ , the higher the parameter $\kappa$ , the more an individual cares about exercise due to the higher reference level of exercise. The dynamic element yields the property that we call the dynamic peer effect. It captures that a higher stock of capital—either built over time due to economic growth or observed at a given point in time among different countries or population subgroups—endogenously increases peer effects :

(6) \begin{equation} \frac{\partial \varepsilon (k)}{\partial k}\gt 0\,. \end{equation}

Tying peer effects to economic development means that over time, as a country develops, individuals become on average more influenced by peers’ exercise behavior, ceteris paribus.

2.1.3. Body weight

The choices of $C$ and X have an impact on body weight change $\dot{W}$ . We connect weight gain to net energy intake, the difference between energy intake and expenditure [described in a general manner by Schofield (Reference Schofield1985)]. Energy intake is a function of food consumption. We introduce a modification in the Schofield equation to take into consideration the role of exercise, work, and sedentary leisure. Recall that the usual Schofield equation is $\dot{W}=\lambda _{C}C-\lambda _{W}W$ , where the parameter $\lambda _{C}\gt 0$ represents the energy density of food (measured in joules per unit of food consumed), and $\lambda _{W}\gt 0$ reflects a metabolic rate (measured in joules per unit of weight). In this expression, calorie expenditure is a fixed proportion $\lambda _{W}$ of body weight $W$ . Implicitly, calorie expenditure is measured for a unit of time of one, which can be one year or one day, and each type of activity during this unit of time exerts the same amount of calories. By contrast, in our model, we take into consideration that individuals allocate one unit of time to activities that exert different amounts of calories. This unit of time is spent in exogenous sedentary leisure $\bar{S}$ , endogenous exercise $X$ , and labor $N$ . Therefore, we rewrite the Schofield equation as

(7) \begin{equation} \dot{W}=\lambda _{C}C-\frac{\left (\lambda _{S}\bar{S}+\lambda _{X}X+\lambda _{N}N\right )}{N}W. \end{equation}

In this expression, $\lambda _{S}\bar{S}$ represents the basal metabolic rate (BMR), which is basic energy expenditure to maintain the functioning of a body at rest. The terms $\lambda _{X}X$ and $\lambda _{N}N$ , with $\lambda _{X}\gt 0$ and $\lambda _{N}\gt 0$ , denote the extent to which time spent on exercise and labor reduces net energy. Note that $\lambda _{C}$ is a rate in front of the variable $C$ . In the same way, the term in front of $W$ is expressed as a rate. For that reason, $\lambda _{S}\bar{S}+\lambda _{X}X+\lambda _{N}N$ is divided by $N$ (since $W$ is expressed in total terms). While sedentary leisure $\bar{S}$ is fixed and proportional to weight, calorie expenditure associated to non-sedentary leisure and work $\lambda _{X}X+\lambda _{N}N$ vary with individuals’ choices. Since $N=1-\bar{S}-X$ , it is straightforward that when individuals choose to exercise more, they spend relatively fewer calories at work.

2.1.4. Preferences and optimal choices

Individuals face a constrained intertemporal optimization problem that is described as follows. Instantaneous utility is given by

(8) \begin{equation} U(C,\hat{X},W)=u(C,\hat{X})-v\left [\left (W-W^{I}\right )^{2}\right ]\,. \end{equation}

Utility positively depends on food consumption and exercise. An exercise reference level $\bar{X}$ increases marginal utility of exercise. Weight is not a choice variable, and conspicuous behavior is solely reflected in individuals’ exercise choices. We account for the fact that disutility is caused by body weight in excess or below a reference weight, $W^{I}\in \mathbb{R}_{+},$ which could, for example, be an ideal weight from a health perspective. There are two ways to interpret this formulation. Considering that individual and average weight are equal in equilibrium, this formulation can denote obesity-related externalities and justifies considering policy interventions discussed in the conclusion. Alternatively, this formulation can denote that individuals do not internalize internalities related to being overweight. Individuals are in that case boundedly rational or “weight unconscious” [e.g. Mathieu-Bolh and Wendner (Reference Mathieu-Bolh and Wendner2020); Mathieu-Bolh (Reference Mathieu-Bolh2021); Yaniv et al. (Reference Yaniv, Rosin and Tobol2009)]. Boundedly rational choices connect to the vast literature on biased health choices that have roots in psychological biases or informational problems [lack of information or cost of acquiring or processing information, e.g. Chetty et al. (Reference Chetty, Looney and Kroft2009)].Footnote ⁴

Assumption 1. The sub-utility function $u\;:\;\mathbb{R}_{+}^{2}\rightarrow \mathbb{R}$ is twice continuously differentiable, increasing in both arguments and strictly concave. Moreover, $\lim _{C\downarrow 0}u_{C}=\lim _{\hat{X}\downarrow 0}u_{\hat{X}}=\infty$ , and $u(C,\hat{X})\leq q\,\rho \,e^{\beta \rho t}$ , where $q\in \mathbb{R}_{+}$ and $\beta \lt 1$ . The sub-utility function $v\;:\;\mathbb{R}_{+}\rightarrow \mathbb{R}$ is twice continuously differentiable, increasing and strictly convex in its argument $(W-W^{I})^{2}$ . Moreover, $v\left [\left (W-W^{I}\right )^{2}\right ]\leq q_{w}\,\rho \,e^{\beta _{w}\rho t}$ , where $q_{w}\in \mathbb{R}_{+}$ and $\beta _{w}\lt 1$ .

Notice that the limit conditions prevent the individual from not consuming or exercising at all, while the latter conditions imply boundedness of intertemporal utility. The boundedness condition on function $v$ is satisfied if $v$ is an increasing function, and there exists a maximum weight.

The intertemporal utility function is

(9) \begin{equation} \int _{t=0}^{\infty }U(C,\hat{X},W)e^{-\rho \tau }d\tau \,, \end{equation}

where $\rho \gt 0$ is the constant rate of time preference. Assumption 1 implies boundedness of the utility integral.Footnote ⁵

Given the DOP, $\varepsilon (\bar{k})$ , individuals choose $C$ and $\hat{X}$ , to maximize (9), subject to their initial endowment of wealth $K_{0}\gt 0$ , their flow budget constraint [combining (4) and (3)]:

(10) \begin{equation} \dot{K}=rK+w\bar{L}-\hat{p}_{X}\varepsilon \left (\bar{k}\right )\bar{X}-p_{C}C-\hat{p}_{X}\hat{X}\,, \end{equation}

and a No-Ponzi-Game (NPG) constraint:

(11) \begin{equation} \underset{\tau \rightarrow \infty }{lim}\:e^{-R(t,\tau )}K\geqslant 0\,, \end{equation}

where $R(t,\tau )=\int _{t}^{\tau }r(v)dv$ represents the interest factor, and:

(12) \begin{equation} \hat{p}_{X}\equiv w+p_{X}. \end{equation}

Price $\hat{p}_{X}$ represents the total cost of effective exercise that includes the cost of exercise expenditure and the opportunity cost of exercise.

We solve the model (see Section A.1) applying Pontryagin’s maximum principle (Pontryagin et al., Reference Pontryagin, Boltyanskii, Gamkrelidze and Mishchenko1962). We deduce the following expressions for the growth rates of food consumption and exercise. Noticing that in equilibrium, average exercise expenditure equals individual’s exercise expenditure, $\bar{X}=X$ , and combining the optimality conditions and the relation between effective and actual exercise (4), we obtain the growth rates of consumption and exercise as (see Section A.2):

(13) \begin{equation} \frac{\dot{C}}{C}=\Omega ^{C}\left (C,\hat{X}\right )(r-\rho )+\Phi ^{C}\left (C,\hat{X}\right )\left (\Delta \frac{\dot{w}}{w}\right )\,, \end{equation}

(14) \begin{equation} \frac{\dot{X}}{X}=\Omega ^{X}\left (C,\hat{X}\right )\left (r-\rho \right )+\frac{\dot{\varepsilon }\left (k\right )}{1-\varepsilon \left (k\right )}-\Phi ^{X}\left (C,\hat{X}\right )\left (\Delta \frac{\dot{w}}{w}\right )\,, \end{equation}

where $0\lt \Delta \equiv \frac{w}{w+p_{X}}\lt 1$ , and $\Omega ^{C}$ , $\Omega ^{X}$ , $\Phi ^{C}$ , $\Phi ^{X}$ are elasticities defined in Section A.2.

The difference between the growth rate of food consumption and the growth rate of exercise expenditure is therefore expressed as

(15) \begin{eqnarray} \frac{\dot{C}}{C}-\frac{\dot{X}}{X}&=&\underbrace{\left [\Omega ^{C}\left (C,\hat{X}\right )-\Omega ^{X}\left (C,\hat{X}\right )\right ]\left (r-\rho \right )}_{\text{ECE}}+\underbrace{\left [\Phi ^{X}\left (C,\hat{X}\right )+\Phi ^{C}\left (C,\hat{X}\right )\right ]\left (\Delta \frac{\dot{w}}{w}\right )}_{\text{ROC}}\nonumber \\[5pt] & & -\underbrace{\frac{\dot{\varepsilon }\left (k\right )}{1-\varepsilon \left (k\right )}}_{\text{DPE}}\,. \end{eqnarray}

The growth rate of food consumption differs from the growth rate of exercise expenditure due to three terms. The first term represents elasticities for consumption and exercise (ECE). It includes two elasticities, respectively, $\Omega ^{C}\left (C,\hat{X}\right )$ and $\Omega ^{X}\left (C,\hat{X}\right )$ . The second term represents the rising opportunity cost of exercise (ROC), and the third term represents the dynamic peer effect (DPE). Note that accounting for the fact that exercise is a consumption expenditure influences the ROC through two channels. One channel operates through $\Phi ^{C}\left (C,\hat{X}\right )\lt 0$ , which denotes that exercise is an expenditure using up resources that cannot be allocated to food consumption. Other things equal, it decreases the growth rate of food consumption and limits the ROC. The other channel operates through $p_{X}$ and lowers $\Delta \equiv \frac{w}{w+p_{X}}$ . Other things equal, it decreases the importance of ROC related to wage growth. The ultimate impact involves general equilibrium effects that will be accounted for in the next sections.

Proposition 1. For general forms of utility functions, under Assumption 1 , the difference between the growth rate of consumption and the growth rate of exercise can be positive or negative as the DPE is an offsetting force to the ROC.

Proof. See Section A.3.

The term ECE is neither specific to our assumption that exercise is both a consumption and a time expenditure nor to our assumption that exercise is affected by peer behavior. By contrast, the terms DPE and ROC are specific to our model and are present for all utility’s functional forms. For that reason, in what follows, we will focus on the roles of DPE and ROC. On their own, those terms cause a wedge between the growth rates of consumption and exercise and generate the pattern for body weight gain. This mechanism is at the core of our results and holds for all forms of utility functions.

It is worth noticing a fundamental difference between the mechanisms driving the income obesity relation in our model compared to Mathieu-Bolh and Wendner (Reference Mathieu-Bolh and Wendner2020). In their contribution, the driver of obesity is an income effect, which would be captured by the term $\Omega ^{C}\left (C,\hat{X}\right )\left (r-\rho \right )$ in (15), competing with a DPE with respect to low calorie food consumption instead of exercise. By contrast, in our model, the income effect is present only for some specific forms of utility functions. For example, the term ECE disappears when utility is a CES function.

2.2. Firms

A unit mass of competitive firms produces a homogeneous output $Y$ . The production process is described by a Cobb–Douglas production function: $Y=F(K,N)=K^{\eta }N^{1-\eta }$ , where $0\lt \eta \lt 1$ denotes the capital elasticity of production. Factors are paid their respective marginal products:

(16) \begin{equation} \frac{\partial F(K,N)}{\partial K}=r+\delta,\quad \frac{\partial F(K,N)}{\partial N}=w\:, \end{equation}

where $\delta \geq 0$ denotes the rate of depreciation. We introduce a normalization (per unit of labor): $y\equiv Y/N$ . Noting that $k=K/N$ , by homogeneity of degree 1, we can write:

(17) \begin{equation} y=F(k,1)\equiv f(k)=k^{\eta }\,,\quad 0\lt \eta \lt 1. \end{equation}

The first-order conditions (16) then become:

(18) \begin{equation} r(k)=f'(k)-\delta \,,\,\,\,\,\,w(k)=f(k)-k\,f'(k)\,. \end{equation}

A linear production process transforms total consumption into food consumption and exercise consumption. Output can be used for either investment $I$ or consumption such that $Y=p_{C}C+p_{X}X+I$ .

2.3. Equilibrium

To study stability and ultimately enable comparisons between economies or population cross sections, we express the dynamic system in per unit of labor terms, such that $c\equiv C/N$ , $x\equiv X/N$ , and $\mathrm{w}\equiv W/N$ . Furthermore, we express the dynamic system as functions of $x$ and $k$ (see Section A.4 for details):

(19) \begin{eqnarray} & \frac{\dot{k}}{k}= & \left [1-x\left [\kappa k-\Phi ^{X}\left (x,k\right )\eta \Delta (k)\right ]\right ]^{-1}\nonumber \\[5pt] & & \left [x\,\Omega ^{X}\left (x,k\right )(f'(k)-\delta -\rho )+\frac{f(k)}{k}-\delta -\frac{1}{k}\left (p_{C}c(x,k)+p_{X}x\right )\right ]\,, \end{eqnarray}

(20) \begin{eqnarray} & \frac{\dot{x}}{x}= & (1+x)\left [\Omega ^{X}\left (x,k\right )\left (f'(k)-\delta -\rho \right )+\left (\kappa k-\Phi ^{X}\left (x,k\right )\Delta (k)\,\eta \right )\frac{\dot{k}}{k}\right ]\,. \end{eqnarray}

The macroeconomic equilibrium gives rise to a dynamic system in two dimensions: $k$ and $x$ . Indeed, the dynamic system is separable in $\mathrm{w}$ , since the dynamic variables $k$ and $x$ affect body weight $\mathrm{w}$ , but $\mathrm{w}$ does not affect $k$ and $x$ , which is to be expected since weight is not a decision variable.Footnote ⁶ The dynamic system in normalized variables is given by (19)–(20): $\dot{k}=\dot{k}(k,x)$ ; $\dot{x}=\dot{x}(k,x)$ .

We separately determine the change in body weight per unit of labor over time and steady-state body weight. The normalized Schofield equation, also expressed with normalized variables, reads (see Section A.5):

(21) \begin{eqnarray} & \frac{\dot{\mathrm{w}}}{\mathrm{w}}= & \lambda _{C}\frac{c(x,k)}{\mathrm{w}}-\left (\lambda _{S}\bar{s}\left (x\right )+\lambda _{X}x+\lambda _{N}\right )+\Omega ^{X}\left (x,k\right )\left (f'(k)-\delta -\rho \right )\nonumber \\[5pt] & & +\left (\kappa k-\Phi ^{X}\left (x,k\right )\eta \Delta (k)\right )\frac{\dot{k}}{k}\,, \end{eqnarray}

where consumption $c(x,k)$ is derived from the intratemporal optimality condition (ratio of (A1) and (A2) in the appendix), and solely depends on $k$ and $x$ , and where $\bar{s}=\frac{\bar{S}}{N}.$ Since $N$ is endogenous, and depends on $X$ , which depends on $x$ , we have $\bar{s}=\bar{s}\left (x\right )$ .

A steady-state equilibrium is an equilibrium for which $\dot{k}=\dot{x}=0$ . Let $k^{*}$ denote the steady-state value of $k$ , and $x^{*}=x\left (k^{*}\right )$ , $w^{*}=w(k^{*})$ , $\hat{p}_{X}^{*}=p_{X}+w^{*}$ , and $c^{*}=c(x^{*},k^{*}).$ Considering the dynamic system (19)–(20), the steady state is described by the following system (see Section A.8):

(22) \begin{equation} k^{*}=f'^{-1}(\delta +\rho )\,; \end{equation}

(23) \begin{equation} x^{*}=\frac{f(k^{*})-\delta k^{*}}{p_{X}+p_{C}c(x^{*},k^{*})/x^{*}}\,, \end{equation}

where $f'^{-1}(.)$ denotes the inverse function of $f'(.)$ . Equation (22) follows from our model equivalent of the Keynes–Ramsey rule [equation (20)]. The term $c(x^{*},k^{*})/x^{*}$ is implicitly given by dividing (A1) by (A2), which gives $U_{C}(C,\hat{X})/U_{\hat{X}}(C,\hat{X})=p_{C}/\hat{p}_{X}(k)$ . In case $u(C,\hat{X})$ is homothetic, the left-hand side is a function of $(C/\hat{X})=c/[x(1-$ $\varepsilon (k)]$ . By strict concavity, an inverse function exists ( $U_{C}/U_{\hat{X}}$ as a function of $C/\hat{X}$ is one to one). In this case, $c^{*}/x^{*}$ is a function of $k^{*}$ , allowing us to express $x^{*}$ explicitly.Footnote ⁷

As a consequence, neither our static nor our dynamic positionality effect (DPE) impact the steady-state level of $k$ . Equation (23) determines the steady-state share of output to be devoted to exercise, which increases with the degree of peer influence $\varepsilon (k^{*})$ . As $f'(k)$ is a strictly monotonous function, there exists only one value $k^{*}$ satisfying (22). Therefore, the steady state $\left (k^{*},x^{*}\right )$ is unique. Similarly to the standard neoclassical growth model, since there is one predetermined and one jump variable, the unique steady state is a saddle point. As a consequence, for low initial levels of capital $k_{0}\gt 0,$ on the stable arm of the saddle, the stock of capital increases monotonically toward the steady state.

Once $x^{*}$ and $c^{*}$ are determined, we deduce the steady-state body weight per unit of labor $\mathrm{w^{*}}$ :

(24) \begin{equation} \mathrm{w^{*}}=\left (\frac{\lambda _{C}c(x^{*},k^{*})}{\lambda _{S}\bar{s}\left (x^{*}\right )+\lambda _{X}x^{*}+\lambda _{N}}\right )\,. \end{equation}

As the dynamic system is separable in $\mathrm{w}$ (that is, $\mathrm{w}$ is affected both by $x$ and $k$ but not vice versa), a stationary state $(x^{*},k^{*})$ does not imply that $\mathrm{w=\mathrm{w}^{*}}.$ However, as can be easily verified, if $\mathrm{\mathrm{w\gtrless \mathrm{w}^{*}}}$ along a stationary state $(x^{*},k^{*})$ , then $\dot{\mathrm{w}}{\lessgtr 0},$ and $\lim _{t\rightarrow \infty }\mathrm{w}=\mathrm{w^{*}}$ . Consequently, the pattern of the dynamics of $c/x$ along transitional paths impacts on the weight change $\dot{\mathrm{w}}$ , though we may experience periods for which $c/x$ declines, while weight is still increasing, and we may experience periods for which $c/x$ is constant while the weight is declining. That is, $\mathrm{w}$ is lagging behind the dynamic pattern of $c/x$ .Footnote ⁸ The behavior of weight, in our model, gives rise to a rich dynamics depending on both the initial weight and the dynamics of the ratio $c/x$ , following the rise in $k$ on the transitional path.

3.1. Dynamic obesity Kuznets curve

In what follows, we show that the two opposing effects, DPE and ROC, can produce a dynamic Kuznets curve pattern for obesity. We also explain the puzzling fact that high-income earners may increase exercise expenditure despite its rising opportunity cost.

Using the functional form (5), in normalized variables, the DPE and ROC are re-expressed as

(25) \begin{equation} \text{DPE}=\frac{\dot{\varepsilon }\left (k\right )}{1-\varepsilon \left (k\right )}=\kappa \dot{k}\,; \end{equation}

(26) \begin{equation} \text{ROC}=\left (\underbrace{\Phi ^{X}+\Phi ^{C}}_{=1}\right )\Delta (k)\frac{\dot{w}}{w}=\frac{1}{\hat{p}_{X}}\left (-kf''(k)\right )\dot{k}\,. \end{equation}

Considering (15), the proof of Proposition 1, and that $\frac{\dot{X}}{X}-\frac{\dot{C}}{C}=\frac{\dot{x}}{x}-\frac{\dot{c}}{c}$ , we have:

(27) \begin{equation} \frac{\dot{c}}{c}-\frac{\dot{x}}{x}=\underbrace{\frac{-kf''(k)}{p_{X}+w}\dot{k}}_{\text{ROC}}-\underbrace{\kappa \dot{k}}_{\text{DPE}}. \end{equation}

Specifically, for $\dot{k}\gt 0$ ,

(28) \begin{equation} \frac{\dot{c}(t)}{c(t)}-\frac{\dot{x}(t)}{x(t)}\gtreqless 0\Longleftrightarrow \frac{-kf''(k)\big )}{p_{X}+w}\gtreqless \kappa \Longleftrightarrow \underbrace{\frac{(1-\eta )\eta }{k^{1-\eta }\left [p_{X}+(1-\eta )k^{\eta }\right ]}}_{\equiv h(k)}\gtreqless \kappa. \end{equation}

We now shed light on the difference between the growth rates of food consumption $c$ and exercise $x$ in equilibrium. To this end, we restrict our study to the case of a growing economy. Recall that $k_{0}$ is the initial capital stock at $t=0$ , and $k^{*}=\left [\eta/\left (\delta +\rho \right )\right ]^{1/(1-\eta )}$ is the steady-state level of capital (as $t\rightarrow \infty$ ).

Proposition 2 shows that this assumption implies that, over time, the sign of the difference $\frac{\dot{c}(t)}{c(t)}-\frac{\dot{x}(t)}{x(t)}$ changes from positive to negative. Assumption 2 restricts the parameters of our model $(k_{0},p_{X},\delta,\eta,\kappa,\rho )$ such that (i) $k_{0}$ is “low," i.e., a higher income raises net-calorie intake and (ii) $k^{*}$ is “high” so that consumption grows by less than effective exercise, i.e., net calorie intake is negative. One way to think about Assumption 2 is that for all points in parameter space $(k_{0},p_{X},\delta,\eta,\rho )$ , there exists a $\kappa \gt 0$ such that Assumption 2 is satisfied.

Below, we show that the pattern of the difference $\frac{\dot{c}(t)}{c(t)}-\frac{\dot{x}(t)}{x(t)}$ along the transitional path impacts the evolution of weight and gives rise to a dynamic obesity Kuznets curve (unless the initial weight is very high).

Assumption 3 poses a natural restriction on the development of weight. It is not the change in exercising, $x$ , (alone) that determines the change in weight, it rather is the change in the ratio of calorie intake to calorie expenditure, $(c/x)$ , that impacts the change in weight. In a growing economy ( $\dot{k}\gt 0$ ), both $c$ and $x$ increase. In light of Proposition 2, for $t\lt \hat{t}$ , the ratio $(c/x)$ increases, and so does the weight. However, for $t\gt \hat{t}$ , the ratio $(c/x)$ decreases, while $x$ increases. The assumption puts an upper bound on the rise of $x$ . Equipped with Assumptions 2 and 3, we can now establish Proposition 3.

Proposition 3 helps us explain the evolution of obesity that goes with economic development as coming from a change in preferences connected to social status. The intuition is that peer effects with respect to exercise are higher in the future than in the present. Consequently, ceteris paribus, the marginal utility of $X$ increases over time. In response, individuals shift $X$ from the present to the future, which implies a higher growth rate of exercise than in the absence of peer effects. For low levels of economic development, Proposition 2 predicts that the growth rate of $c$ exceeds the growth rate of $x$ (note that $\frac{\dot{X}}{X}-\frac{\dot{C}}{C}=\frac{\dot{x}}{x}-\frac{\dot{c}}{c}$ ). As a consequence, the ratio $c/x$ and body weight increase, as shown by Proposition 3. At some point of economic development $\hat{t}$ , as the stock of capital per unit of labor exceeds a certain threshold $\hat{k}$ , Proposition 2 predicts that the growth rate of $x$ exceeds the growth rate of $c$ . As a consequence, the ratio $c/x$ decreases, as shown in Proposition 3. Since the evolution of body weight is tied to the evolution of $c/x$ , even if lagged, it is easy to show that the growth rate of body weight eventually decreases. Therefore, Propositions 2 and 3 provide an explanation for the empirically estimated dynamic obesity Kuznets curve.

More specifically, Proposition 3 requires $W(0)$ not to be too high. The intuition is obtained from the Schofield equation. If $W(0)$ is too high relative to $C$ and $X$ , there exists an initial period for which $\dot{W}\lt 0$ . Next, suppose that $W(0)$ is less than this level. Then, for $t\in \left [t_{0},\hat{t}\right ]$ , $\dot{W}\gt 0$ , as $(c/x)$ increases in this period and so does the weight term under Assumption 3. For $t\gt \hat{t}$ , weight continues to increase until, at $t=\hat{t}_{w}$ , the ratio $(c/x)$ becomes so small that the Schofield equation requires weight to decrease. If $W(t)\gt W^{*}$ along a (roughly) stationary state $(x^{*},k^{*})$ , $\dot{W}\lt 0$ , and weight asymptotically converges to its steady-state level. Assuming that the initial weight is not too high is consistent with empirical observations that average body weight is relatively low in counties with low levels of economic development compared to countries with high levels of economic development.

A consequence of Corollary 1 is that without the DPE, the ratio $c/x$ and body weight would never decrease. This would yield two counterfactual results. It would imply that exercise expenses always grow at a lower pace than food consumption, and that obesity always rises and never exhibits a Kuznets curve along a transitional path.Footnote ¹⁰

It is not possible to explicitly isolate the effect of the ROC on the results because for two reasons. First, one ingredient of the ROC is the equilibrium wage. The equilibrium wage rises endogenously as capital accumulates. It is therefore not possible in our model to eliminate the rise in the opportunity cost of leisure. Second, the other ingredient of the ROC is the price $p_{X}$ that individuals pay for leisure. If we were to set $p_{X}=0$ , this would decrease the relative cost of exercise and the coefficient $\Delta$ , directly increasing the ROC ceteris paribus. It would also modify the optimal choice of exercise relative to food consumption and therefore influence capital accumulation, and the marginal product of labor, which indirectly influences both DPE and ROC. Therefore, it is not possible to derive analytical results from setting $p_{X}=0$ . However, we study the effects of relaxing the assumption of $p_{X}\gt 0$ on the dynamic Kuznets curve in the numerical section.

3.2. Static obesity Kuznets curve

In what follows, we draw a parallel between mechanisms operating for the dynamic obesity Kuznets curve and the static obesity Kuznets curve. The following comparative static analysis can be considered a cross-sectional analysis, comparing countries or individuals with different steady-state wealths $k^{*}$ . It can also be considered a comparative static analysis of a single country for which a change in a technology or preference parameter causes a change in $k^{*}$ . In contrast to the dynamic Kuznets curve, we study how steady-state weight varies with different steady-state values of $k$ .

To compare steady states, we totally differentiate expression $c^{*}=\alpha/(1-\alpha )\left (\hat{p}_{X}^{*}/p_{C}\right )\left (1-\varepsilon (k^{*})\right )x^{*}$ with respect to $k^{*}$ , so we study the DPE and ROC associated with $dk^{*}\gt 0$ (see Section A.9 for details):

\begin{equation*} ROC^{*}=\Delta \left (k^{*}\right )\frac {dw^{*}}{w^{*}}=h(k^{*})\,, \end{equation*}

\begin{equation*} DPE^{*}=\frac {d\varepsilon (k^{*})}{\left (1-\varepsilon (k^{*})\right )}=\kappa \,, \end{equation*}

where $dw^{*}=\partial w/\partial k\mid _{k=k^{*}}dk^{*}$ , and $d\varepsilon (k^{*})=\partial \varepsilon/\partial k\mid _{k=k^{*}}dk^{*}$ . The difference between the change of $c^{*}$ and the change of $x^{*}$ upon a rise in $k^{*}$ is given by

(29) \begin{eqnarray} \frac{dc^{*}/c^{*}-dx^{*}/x^{*}}{dk^{*}/k^{*}}=\frac{1}{dk^{*}/k^{*}} \left [\underbrace{h(k^{*})}_{\text{ROC}^{*}}-\underbrace{\kappa }_{\text{DPE}^{*}}\right ]\,, \end{eqnarray}

where $\text{ROC}^{*}$ represents the higher opportunity cost of time spent on exercise that goes with a higher marginal product of labor associated with a higher steady-state capital stock, and $\text{DPE}^{*}$ represents the dynamic peer effect that denotes the increase in peer effects associated with a higher steady-state capital stock. The terms $\text{ROC}^{*}$ and $\text{DPE}^{*}$ are the steady-state equivalents of the terms ROC and DPE presented in the dynamic setting.

For our functional specification, $k^{*}=\left [\eta/\left (\delta +\rho \right )\right ]^{1/(1-\eta )},$ with $0\lt \delta,\eta,\rho \lt 1$ . We make the following assumption:

Assumption 4 plausibly implies that $k^{*}(\delta,\eta,\rho )$ is an increasing function of the output elasticity $\eta$ . Specifically, $k^{*}(\delta,0,\rho )=0$ , $\partial k^{*}/\partial \eta \gt 0$ , and $\lim _{\eta \uparrow 1} k^{*} =\infty$ . Thus, under Assumption 4, $k^{*}\in (0,\infty )$ for $\eta \in (0,1)$ . Now, consider $h(k^{*})$ . We already established that $h'(k^{*})\lt 0$ . It is easy to verify that $\lim _{k^{*}\downarrow 0}h(k^{*})=\infty$ . Likewise, $\lim _{k^{*}\uparrow \infty }h(k^{*})=0$ . Thus, under Assumption 3, for $k^{*}\in (0,\infty )$ , $h(k^{*})\in (0,\infty )$ .

As $k^{*}=k^{*}(\delta,\eta,\rho )$ , we know from (28) that $h(k^{*})=h^{*}(p_{X,}\delta,\eta,\rho ).$ As $\eta$ varies between $(0,1)$ , $h^{*}$ varies between $(0,\infty ).$ However, $h^{*}$ also declines in $p_{X}$ and rises in $(\delta +\rho )$ . Thus, for every $\kappa \gt 0$ , there always exist parameter combinations for which $h^{*}(p_{X,}\delta,\eta,\rho )=\kappa$ . This parametric condition implies a specific steady-state level of capital, $\hat{k}^{*}$ . Clearly, for $k^{*}\lt \hat{k}^{*}$ , $h^{*}\gt \kappa$ , and for $k^{*}\gt \hat{k}^{*}$ , $h^{*}\lt \kappa$ . From (29), we deduce Proposition 4, which is the steady-state equivalent of Proposition 2.

Proposition 4 shows that the sign of $\text{ROC}^{*}-\text{DPE}^{*}$ becomes negative beyond a certain threshold of steady-state level of capital per worker. The result very much depends on the behavior of $\text{ROC}^{*},$ as $\text{DPE}^{*}$ is a constant. We note that $\text{ROC}^{*}$ depends negatively on $p_{X}$ , as the resource opportunity cost (foregone wages times the share of $w$ to $(p_{X}+w)$ ) declines in the “fixed cost” $p_{X}$ .

The corollary shows that the higher $p_{X}$ the lower $\hat{k}^{*}$ and the larger the range of $k^{*}$ for which $\frac{dx^{*}}{x^{*}}\gt \frac{dc^{*}}{c^{*}}$ . A necessary condition for a static obesity Kuznets curve to occur is $\frac{dx^{*}}{x^{*}}\gt \frac{dc^{*}}{c^{*}}$ . Therefore, taking into account that exercise is a type of consumption expenditure (i.e., $p_{X}\gt 0$ rather than $p_{X}=0$ ) shrinks the space $(k_{0},\hat{k}^{*})$ for which $\frac{dx^{*}}{x^{*}}\lt \frac{dc^{*}}{c^{*}}$ and widens the range of $k^{*}$ for which a static Kuznets curve is obtained.

The sign of $(h(k^{*})-\kappa )$ impacts the steady-state weights across sections, as $\frac{dx^{*}}{x^{*}}\gt \frac{dc^{*}}{c^{*}}$ is a necessary condition for the weight to decline.

The comparison of body weight for different steady-state wealth levels is obtained by totally deriving body weight (see (A16) in the appendix) with respect to $k^{*}$ , which yields:

(30) \begin{equation} \frac{d\mathrm{w^{*}}}{dk^{*}}=\frac{\lambda _{C}c^{*}/x^{*}}{z^{*}}\left [\frac{\frac{d\left (c^{*}/x^{*}\right )}{c^{*}/x^{*}}-\frac{dz^{*}}{z^{*}}}{dk^{*}}\right ] \end{equation}

where $z\left (k^{*}\right )=\lambda _{S}\bar{s}\left (x^{*}\right )/x^{*}+\lambda _{X}+\lambda _{N}/x^{*}$ represents total calorie expenditure divided by exercise expenditure. Expressed in terms of elasticities, we obtain:

\begin{equation*} \frac {d\mathrm {w^{*}/w^{*}}}{dk^{*}/k^{*}}=\left [\frac {\frac {d\left (c^{*}/x^{*}\right )}{c^{*}/x^{*}}-\frac {dz^{*}}{z^{*}}}{dk^{*}/k^{*}}\right ]\,, \end{equation*}

where we consider that $\frac{\lambda _{C}c^{*}/x^{*}}{z^{*}}=\mathrm{w^{*}}.$

The first term on the right-hand side in the numerator of (30) represents the percentage change of net calorie intake. The second term is the percentage change of total calorie expenditure relative to exercise expenditure. When exercise time $x^{*}$ increases, $dz^{*}\lt 0$ . Thus, a necessary condition for steady-state weight to decrease across sections is $d\left (c^{*}/x^{*}\right )/dk^{*}\lt 0$ .

As we consider $dk^{*}\gt 0,$ our model implies $d\mathrm{w^{*}}/dk^{*}\gt 0$ for $k^{*}\lt \hat{k}^{*}$ . However, for $k^{*}\gt \hat{k}^{*}$ , our model implies two opposing effects on the steady-state weight. For $k^{*}\gt \hat{k}^{*}$ , $(c^{*}/x^{*})$ decreases for $dk^{*}\gt 0$ . At the same time, $z^{*}$ also decreases. To gain more insight, we rewrite the above total derivative as follows:

\begin{equation*} \frac {d\mathrm {w^{*}/w^{*}}}{dk^{*}/k^{*}}=\left [\frac {\left (h(k^{*})-\kappa \right )-dz^{*}/z^{*}}{dk^{*}/k^{*}}\right ]\,. \end{equation*}

Intuitively, recall that the steady state $k^{*}\in \mathbb{R}_{+}$ (by, e.g., varying parameter $\eta$ ). As $\left (h(k^{*})-\kappa \right )\gt 0$ for $k^{*}\lt \hat{k}^{*}$ by definition, and $dz^{*}/z^{*}\lt 0$ , condition (i) of the assumption can only be satisfied for a $\tilde{k}^{*}\gt \hat{k}^{*}$ . This condition also excludes the case for which $-dz^{*}/z^{*}\gt \kappa$ for all $k^{*}$ . It requires that the degree of peer influence is large enough. Condition (ii) of the assumption imposes a monotonicity condition: $h(k^{*})-\kappa$ is required to decline by more than $dz^{*}/z^{*}$ for $k^{*}\gt \tilde{k}^{*}.$ In this case $d\mathrm{w^{*}}/dk^{*}\lt 0$ for all $k^{*}\gt \tilde{k}^{*}.$ While Assumption 5 might sound restrictive at first sight, it is always satisfied for the standard utility specifications employed in the numerical simulations (see below).

If $\kappa =0$ , then no static obesity Kuznets occurs. The $\text{DPE}^{*}$ generates substitution toward exercise and drives the negative correlation between the stock of capital and body weight for high values of the steady-state capital stock. In the absence of the $\text{DPE}^{*}$ , for high values of the steady-state capital stock, the main effect is the $\text{ROC}^{*}$ that unambiguously results in a decrease of exercise and a higher steady-state body weight. Therefore, the introduction of dynamic peer effects helps explain why, despite a higher opportunity cost of exercise, rich individuals may exercise more than poor individuals.

With Proposition 3, we showed that the evolution of the stock of capital over time generates ROC and DPE, eventually yielding a dynamic obesity Kuznets curve. With Proposition 4, we explain how differences in $\text{ROC}^{*}$ and $\text{DPE}^{*}$ are tied to different steady-state capital stocks between poor and rich countries (or individuals). Propositions 4 and 5 are consistent with evidence presented in the introduction that $d\mathrm{w^{*}}/dk^{*}\gt 0$ for poor countries (or individuals) and $d\mathrm{w^{*}}/dk^{*}\lt 0$ for rich countries (or individuals) and suggests that dynamic peer effects associated with the consumption of time consuming goods plays a larger role for rich than poor countries (or individuals).

3.3. Calorie consciousness

In this section, we study the behavior of individuals who are both weight conscious and calorie conscious. Individuals internalize the net effect of calorie intake on weight gain [equation (7)], accounting for the fact that labor and exercise choices are endogenous [equation (1)] and that exercise choices are influenced by peer effects [equation (4)]. The formulation of the problem is presented in Section A.12.

As expected, calorie consciousness reinforces the choice of exercise over consumption and results in lower equilibrium body weight. To understand the assumption and gain intuition, consider an individual for whom $W\gt W^{I}$ , that is, for whom a gain in weight reduces utility (parallel arguments apply for the opposite case). Notice that the term $\frac{W}{N}(\lambda _{N}-\lambda _{X})$ describes the weight change caused by an increase of $\hat{X}$ by one unit, as seen in (A18). While we typically expect this term to be negative, in principal it can take on either positive or negative values. The term $\lambda _{C}$ describes the weight gain caused by an increase of $C$ by one unit. Shifting expenses by reducing $C$ by one dollar and increasing $\hat{X}$ by one dollar results in a reduction of $C$ by $1/p_{C}$ units and an increase of $\hat{X}$ by $1/\hat{p}_{X}$ units. Thus, the assumption in Proposition 6 necessitates that the weight loss resulting from reducing $C$ by one dollar must surpass the weight change incurred by increasing $\hat{X}$ by one dollar.Footnote ¹¹ Weight consciousness leads to an increase in utility through this shift in expenses, consequently lowering the value of $C/\hat{X}$ . With calorie consciousness $C/\hat{X}$ is consistently lower for every $k$ compared to when calorie consciousness is absent. This also holds true for the steady state: since calorie consciousness does not impact the steady-state capital stock [as seen in equation (A21)], and $C/\hat{X}$ is lower, the steady-state weight is consequently reduced under calorie consciousness.

The Kuznets curve continues to reflect the competing rise in opportunity cost of exercise and peer effects. Calorie consciousness may expedite the transition from a positive to a negative income–obesity relationship or push the Kuznets curve downward.

4.1. Calibration

First, we set parameters for the Schofield equation. We use data on time use, calories spent exercising, and BMR to calibrate the parameters of the Schofield equation.

An individual spends about 2000 hours per year working (40 hours times 50 weeks) out of 8736 total hours in a year (24 hours time 7 days times 52 weeks). The time that is not spent working represents 6736 hours of leisure time (8736 − 2000 = 6736) and is split between sedentary leisure and exercise. With exercise time representing only 122 hours (20 minutes times 365 days divided by 60), sedentary leisure represents 6614 hours (6736 − 122 = 6614). As a result, sedentary leisure represents a fraction equal to $\bar{S}=$ 6614/8736 = 75.7% of total time, and time that is not spent on sedentary leisure represents a fraction of $\bar{L}=1-0.757=24.3\%$ of total time. This time is split between exercise, which represents a fraction equal to 122/8736 = 1.4% of total time, and work, which represents a fraction N = 2000/8736 = 22.9% of total time. The average of men and women average weight in the USA is 185 pounds. On average, an individual at rest spends $\lambda _{S}\bar{S}W=$ 1577.5 calories per day. As a result, $\lambda _{S}=1577.5/(0.757*185)=11.26$ .

Based on the yearly American time use survey by the Bureau of Labor Statistics (2023), time spent “participating in sports, exercise and recreation," measured since 2003, represents 0.33 hours (20 minutes) a day for the average individual. How many calories do those activities burn? The 2008 Bureau of Labor Statistics Spotlight on Statistics indicates that the three most popular types of exercise are walking, weightlifting, and using cardiovascular equipment (see Figure A.13.2 in the appendix). For a 185 pound individual, we estimate that these activities, respectively, burn 2.0, 1.4, and 4.7 calories per pound per hour. The calories burnt are taken from the chart provided by Harvard Medical School (2021) and activities specifically correspond to walking 4 miles per hour, general weight lifting, and high impact step (or vigorous rowing). We estimate that calories burnt by an individual splitting their exercise time among those three activities, weighted by their popularity, equals 2.5 calories per pound per hour. The details of our calculation are provided in Table A.13.1 in the appendix. Thus, an average individual of 185 pounds spends $2.5*185=462.5$ calories per hour. Since individuals exercise 20 minutes a day, they spend $\lambda _{X}XW=$ 154 calories per day. As a result, $\lambda _{X}=154/(0.014*185)=59.5$ .

Based on the U.S. Department of Health and Human Services and U.S. Department of Agriculture (2015), we use the average of daily caloric expenditure of men and women, which equals $((1600+2400)/2+(2000+3000)/2)/2=2250$ . With 1577.5 calories spent in sedentary leisure and 154 calories spent exercising, calories spent at work are $\lambda _{N}NW=2250-1577.5-154=518.5$ . As a result, $\lambda _{N}=518.5/(0.23*185)=12.2$ .

In order to estimate parameter $\lambda _{C}$ (energy density of food), we use the Schofield equation, written for a stationary body weight: $\lambda _{C}=\frac{\left (\lambda _{S}\bar{S}+\lambda _{X}X+\lambda _{N}N\right )W/N}{C}=\frac{2250}{4}=562.5$ per unit of food expenditure per day. As explained above, the numerator represents total calorie expenditure per day. The denominator represents the quantity of food consumed per day, which is between 3 and 5 pounds. We take 4 pounds for the baseline calibration. As a result, parameter $\lambda _{C}$ represents energy intake per unit of food expenditure per day. Note that there is no distinction between men and women or individuals of different ages within households in the per quintile food consumption data.

We need a consistent estimate for food and exercise, so we use cost per day. To estimate the price of exercise, we use the work by Valero-Elizondo et al. (Reference Valero-Elizondo, Salami, Osondu, Ogunmoroti, Arrieta, Spatz, Younus, Rana, Virani, Blankstein, Blaha, Veledar and Nasir2016), who estimate the marginal benefit of exercising (which, in equilibrium, equals the marginal cost) equal to $2500 per year. We divide it by 365 days and by the time a person spends exercising daily to obtain $p_{X}$ , the price per day. Note that the marginal benefit (or cost) of exercising is high as it represents more than the sole expenditure on exercise-related activities or goods in that case. It also encompasses additional benefits in the form of savings on health care expenditure due to better health outcomes. So it overstates the direct cost of goods and services connected to calorie expenditure.

The average amount spent on food since 2003 is $2,251 per person year. We obtain this number as follows. Based on the Consumer Expenditure Survey, aggregate food consumption expenditure per year equals 776,647 millions of dollars. The share of aggregate food expenditure coming from middle income households is 17.8%. The average number of consumer units in the middle-income category is 24,560 thousands and the number of person per consumer unit is 2.5. As a result, 776,647,000,000 * 17.8%/24,560,000/2.5 = $2,251 per year per person. We divide this number by 365 days and by the quantity of food a person consumes in a day (4 pounds) to obtain $p_{C}$ , the price per pounds.

The model is simulated using a production function with a unit elasticity of substitution between consumption and effective exercise ( $\zeta =0$ ) and a unit intertemporal elasticity of substitution ( $\gamma =1$ ). We estimate the remaining yearly parameters. They are consistent with the standard range of parameters of theoretical growth models [cf. Turnovsky (Reference Turnovsky2000); Barro and Sala-i-Martin (Reference Barro and Sala-i-Martin2003)] and are adjusted to reproduce some essential features of the US economy. We set the production elasticity of capital $\eta$ , the rate of depreciation $\delta$ , and the rate of time preference $\rho$ to obtain a capital output ratio of 3 [e.g. Burda and Wyplosz (Reference Burda and Wyplosz2017, p. 64)].

Parameter $\alpha$ , which represents the taste for consumption relative to leisure, and parameter $\kappa$ , which represents the exogenous component of the degree of peer influence modify steady-state peer effects, consumption, calorie expenditure, and body weight. However, neither are directly observable. We set those parameters to reflect the following empirical observations. Our calibration generates a degree of peer influence close to 0.4, which is consistent with experimental estimates,Footnote ¹³ and a ratio of exercise related expenditure over food expenditure close to one.

Additionally, the food consumption-income ratio is 10% in the USA, and total consumption in output is about 65%. Because our model has only one good, which is food consumption, if we set parameters to reproduce a consumption output ratio of 10%, it would mean that most resources are allocated to investment. The economy and food consumption would grow too fast compared to reality.

At the same time, if we set parameters to reproduce a consumption output ratio of 65%, food consumption and its effect on body weight would largely be overstated. With our parameters, we obtain a ratio of for food consumption in total output between 10 and 65%. Parameter values are summarized in Table 2.

Table 2. Parameters

Table 3 shows the fit between actual and simulated steady-state economies.

Table 3. Actual and simulated economies

4.2. Baseline scenario

First, we build the curve that connects steady-state body weight to the steady-state capital stock. Recall that in our model, peer effects have an endogenous component tied to the level of capital per unit of labor. In the steady state, the stock of capital is given by exogenous parameters, which are the rates of time preference, depreciation, and the production elasticity of capital. These parameters have an impact on the steady-state capital stock $k^{*}$ and steady-state weight. We simulate the steady-state weights by directly changing $k^{*}$ , going from the baseline steady-state capital stock to two times its value.Footnote ¹⁴ Since peer effects also have an exogenous component $\kappa$ related to idiosyncratic differences between individuals or countries, we provide simulations for three different levels of $\kappa$ , at the baseline value of 0.10 and close to it. The static relation between obesity and the stock of capital per unit of labor is presented on the left graph in Figure 1. Second, we build the curve showing the evolution of body weight over time toward the current steady state. The dynamic relation between obesity and the stock of capital per unit of labor is presented on the right graph in Figure 1.

Figure 1. Body weight and capital stock with different degrees of peer influence.

The current steady-state body weight for the USA is 185 pounds, which is the starting point for the static relation between body weight and the stock of capital per unit of labor in our baseline scenario. We find the existence of a static Kuznets curve: for the baseline level of $\kappa$ , the steady-state level of average body weight increases with the average stock of capital up to a level of 186.5 pounds, corresponding to a stock of capital per unit of labor 25% higher than its the baseline and decreases thereafter. The first interpretation of this finding is that the US economy’s tipping point of the Kuznets curve is not yet reached. It will be reached once the stock of capital per unit of labor is 25% higher than its baseline in the simulations.

The second interpretation of this result is that the steady-state average body weight starts being inversely related to wealth when individual wealth is 25% above the average wealth in the USA and is positively related to wealth below this threshold. Furthermore, when the parameter $\kappa$ is higher than in the baseline, DPE* is relatively stronger than ROC*. As a consequence, as expected, steady-state body weight is lower, and the obesity Kuznets curve is even more concave as individuals choose to exercise more. Thus, our simulations also show that exogenous differences in the degree of peer influence between individuals or countries may yield different Kuznets curves.

By contrast, the dynamic evolution of body weight towards its steady-state value of 185 points does not show a dynamic Kuznets curve pattern for body weight in our baseline scenario. It means that given the level of $\kappa$ , the model shows that the increase in the capital stock has so far generated a DPE that has resulted in slowing down the growth rate of body weight but that it has not been sufficient to produce a dynamic obesity Kuznets curve. For both the static and the dynamic relations, the higher $\kappa$ , the more prevalent peer effects, the lower the level of steady-state body weight.

The existence of a static Kuznets curve is consistent with results of empirical studies for the USA presented in the introduction. Given the exogenous degree of peer influence $\kappa$ , the stock of capital per unit of labor would need to be 25% higher for the DPE* to be large enough and steady-state body weight to decrease. The absence of the Kuznets curve to this date is also consistent with US data showing that obesity has increased over time and its growth rate has slowed down (see Figure A.13.3 in the appendix). The DPE may explain the slowdown of the evolution of obesity in the USA as individuals who are influenced by their peers have allocated more time toward exercise as the economy developed. However, this influence has so far been insufficient on its own to produce a decrease in average weight gain.

4.3. Sensitivity analysis

In this subsection, we first show the effect of considering exercise as a consumption expenditure on the static and dynamic Kuznets curves by varying the cost of exercise expenditure, the elasticity of substitution between consumption and effective exercise, the intertemporal elasticity of substitution, and the degree of peer influence.

4.3.1. Exercise as a consumption expenditure

We simulate the static and dynamic relations between body weight and the stock of capital per unit of labor with the price of consumption expenditure ranging from 0 to 20.76. The results are presented in Figure 2.

Figure 2. Body weight and capital stock with different consumer prices for exercise.

When the price of exercise expenditure decreases, the static and dynamic Kuznets curves shift down. The reason is that a lower price of exercise lowers ROC* thereby encouraging exercise. The increase in calorie expenditure results in lower body weight for all levels of the stock of capital.

4.3.2. Elasticity of substitution between consumption and effective exercise

Recall that the baseline scenario corresponds to a unit elasticity, with parameter $\zeta =0$ . We simulate the static and dynamic relations between body weight and the stock of capital per unit of labor with elasticities of substitution between consumption and effective leisure ranging from 0.66 to 2 (corresponding to $\zeta$ ranging from −0.5 to 0.5). The results are presented in Figure 3.

Figure 3. Body weight and capital stock with different elasticities of substitution between food consumption and effective exercise.

When the elasticity of substitution between food consumption and effective exercise becomes larger than in the baseline scenario and equal to 2, with $\zeta =0.5$ , then, as the stock of capital becomes higher in the steady state, individuals substitute food consumption for exercise more easily than in the baseline with unit elasticity of substitution. As a consequence, the relation between weight and the capital stock is decreasing as the steady-state stock of capital becomes higher. In this case, the relation between steady-state weight and the capital stock is not a Kuznets curve. In other words, the DPE* dominates the ROC* for all steady-state levels of capital. In contrast, when the elasticity of substitution between food consumption and effective exercise becomes smaller than in the baseline scenario and equal to 0.66, with $\zeta =-0.5$ , individuals substitute food consumption for exercise less easily as the stock of capital becomes higher in the steady state. The relation between steady-state weight and the capital stock is a Kuznets curve. In this case, until a certain level of steady-state stock of capital per unit of labor, the ROC* dominates and weight increases. Beyond this level, the DPE* dominates, and weight decreases as the steady-state stock of capital becomes higher. The static Kuznets curve is also present in the baseline case (solid curve) but less pronounced than for the lower CES with $\zeta =-0.5$ .

The dynamic relation between weight and the capital stock does not exhibit a Kuznets curve pattern for a wide range of elasticities of substitution between food consumption and effective exercise. For both the static and dynamic relations, the more food consumption and effective exercise are complements, the more prevalent peer effects, the lower the level of steady-state body weight.

To our knowledge, there is no empirical estimate of the elasticity of substitution between food consumption and exercise. If food consumption and exercise are substitutes, we should see a decrease in steady-state weight happening much earlier that in our baseline. If we consider that when people exercise, they also eat more, then food consumption and effective exercise may be complements and the relation between steady-state weight and stock of capital per unit of labor could exhibit a Kuznets curve. However, the model predicts that with an elasticity of substitution of 0.66, the tipping point of the Kuznets curve would happen at a steady-state stock of capital 67% higher and a body weight of 344 pounds, which is a more pessimistic scenario than the baseline scenario.

4.3.3. Intertemporal elasticity of substitution

Recall that the baseline scenario corresponds to an intertemporal elasticity of substitution equal to one with parameter $\gamma =1$ . We simulate the relation between body weight and the stock of capital per unit of labor with intertemporal elasticities of substitution ranging from 0.33 to 1 (corresponding to $\gamma$ ranging from $3$ to one). By definition, the intertemporal elasticity of substitution does not affect the steady state, but it modifies the dynamic evolution of weight as the stock of capital builds up. The results are presented in Figure 4.

Figure 4. Dynamic of body weight with different intertemporal elasticities of substitution.

Figure 5. Dynamic of body weight gain with low and high degrees of peer influence.

When the intertemporal elasticity of substitution becomes smaller, from 1 to 0.33 (as $\gamma$ goes from 1 to 3), the dynamic relation between weight and the stock of capital per unit of labor becomes flatter. As $\gamma$ increases, the effect of the ROC on the difference between the growth rate of exercise and the growth rate of food consumption, $\frac{\dot{x}}{x}-\frac{\dot{c}}{c}$ , becomes smaller, and the effect of the DPE becomes dominant, explaining that the relation between obesity and capital becomes less and less positive. As the DPE become dominant, individuals postpone net calorie intake to the future, which flattens the evolution of body weight.