Taxes on Unhealthy Food and Externalities in the Parental Choice of Children’s Diet

This paper addresses the question whether taxes on unhealthy food are suitable for internalizing intergenerational externalities inflicted by parents when they decide on their children’s diet. Within an OLG model with an imperfectly altruistic parent, the optimal steady state tax rate on unhealthy food is strictly positive. However, it is only second best since it not only reduces food consumption of the child but also distorts the parent's food consumption. Surprisingly, the optimal tax may under- or overinternalize the marginal damage.


Introduction
This short paper addresses the question whether taxes on unhealthy food (fat, sugar, soda, . . .) are suitable for internalizing intergenerational externalities inflicted by parents when they decide on their children's diet. According to OECD (2017), one out of six children is overweight or obese, implying a higher risk of obesity in adulthood and related noncommunicable -often chronic -diseases like, e.g., diabetes. Parents have a large impact on their children's diet, and they are often not perfectly altruistic visa-vis their children. The most direct evidence is provided by Bruhin and Winkelmann (2009) who study how the happiness of children impacts the utility of their parents and estimate that only 21% to 27% of the parents are altruistic. Hence, in choosing their children's diet, a large part of parents do not fully take into account the future health costs of their children and, thus, inflict an intergenerational externality.
We investigate this externality in an overlapping generations (OLG) model of a family in which the parent chooses both its own and its child's diet. Food consumption in childhood increases weight and health costs in adulthood. It also creates habits that raise the marginal utility of food consumption in adulthood. The parent is imperfectly altruistic and takes into account only a part of the child's future utility and health costs.
We find that the optimal steady state tax rate on unhealthy food is indeed strictly positive. However, it is only second best since it not only reduces food consumption of the child but also distorts the parent's food consumption, which is not associated with an externality. Surprisingly, the optimal tax rate may under-or overinternalize the marginal damage. A tax rate increase in a given period reduces the parent's food consumption in this period and thereby gives rise to underinternalization. However, the corresponding fall in the child's consumption ceteris paribus reduces weight in adulthood such that parent consumption in the next period may go up. If this effect is large enough, the optimal tax overinternalizes the marginal damage.
In the economic literature, taxes on unhealthy food (or sin taxes in general) are justified with self control problems of individuals Rabin, 2003, 2006), misperceived health costs (Cremer et al., 2016) or negative cost externalities through health insurance (Allcott et al., 2019). In contrast to our paper, however, the previous literature largely ignores intergenerational externalities between parents and children.
A remarkable exception is the study of Goulão and Pérez-Barahona (2014). They model a family where parents choose an unhealthy activity (e.g., food consumption) that influences health capital, which is later inherited by their children. Parents are not altruistic and do not take the children's future utility into account when deciding on the unhealthy activity. The optimal tax on the unhealthy activity is strictly positive, as in our paper. However, in their analysis the optimal tax is always first best since they do not include a second margin which the tax erroneously distorts. Hence, the important contribution of our paper is to highlight that the parent food consumption is such a second margin which renders the tax on unhealthy food only second best.

Model
Consider an OLG model of a representative family. 1 In each period, the family consists of a parent and a child and each individual lives two periods, childhood and adulthood.
In period t ∈ {0, 1, 2, . . .}, the child's utility from consuming x c t units of food reads The child's consumption is not chosen by the child itself, but by its parent. The parent in period t, in turn, receives consumption utility z p t +V p (x p t , s t ) from the consumption of a numerairé good z p t and own food consumption x p t . Utility of food consumption V p (x p t , s t ) is influenced by habits defined as Hence, habits equal food consumption during childhood. In addition, the parent in period t has to bear health costs C(q t ) that are positively correlated with weight The parent's weight equals food consumption during adulthood plus a share γ ∈ [0, 1] of food consumption during childhood. The net utility of the parent in t reads The utility function V p exhibits positive and declining marginal utility of food con- The marginal utility of habits is assumed to be negative and declining, so V p s (x p t , s t ) < 0 and V p ss (x p t , s t ) < 0. Moreover, the utility function satisfies V p xs (x p t , s t ) > 0. Hence, the parent's marginal utility of food consumption is increasing in past food consumption and ceteris paribus gives the adult incentives to consume more when it has eaten more during childhood. 2 Finally, the marginal health costs are positive and increasing, i.e., C q (q t ) > 0 and C qq (q t ) ≥ 0.
Long-term utility of the parent in period t equals where W t+1 is long-term utility of a child born in period t when it becomes a parent in period t + 1. The weight α ∈ [0, 1] determines the degree of (intergenerational) altruism. If α = 1, the parent takes fully into account the long-term utility of its child and is perfectly altruistic. For α = 0, the parent is non-altruistic. If α ∈]0, 1[, the parent is imperfectly altruistic. Lastly, the family's budget constraint in period t reads where e is a given income, t represents a lump sum transfer received from the government and τ t is the tax rate on food consumption. For simplicity, we subsequently refer to this tax as fat tax. All producer prices are normalized to unity.

Consumption choice of the parent
In period t, the parent chooses its own consumption z p t and x p t as well as the child's consumption x c t in order to maximize utility (5) subject to (1)-(4) and the budget constraint (6) for period t and all periods thereafter. In so doing, the parent in period t takes as given habits s t = x c t−1 . The first-order conditions are (see Appendix A) where the asterisk indicates optimal values. According to (7), the parent chooses own consumption such that the net marginal utility, V p x − 1, equals the sum of marginal health costs, C q , and the fat tax, τ t . Equation (8) states that the parent sets the child's consumption where the net marginal utility, V c x − 1, equals the perceived longterm marginal costs, α(γC q − V p s ) > 0, plus the fat tax, τ t . Hence, in case of a zero tax 2 These properties of the utility function with respect to habits are satisfied for the most commonly used specifications of habits, namely the subtractive habit specification V p (x, s) = v(x − θs) with v > 0 > v and θ ∈]0, 1[, see Lahiri and Puhakka (1998) and Carroll (2000), and the multiplicative Abel (1990). τ t = 0, the parent takes into account only a part of the child's future costs and creates an externality reflected by the share of marginal costs that it ignores, (1−α)(γC q −V p s ). Lagging equations (7) by one period yields For each period t ∈ {0, 1, 2, . . .}, (8) and (9) form a system of equations that determines child consumption in period t and parent consumption in period t + 1 as functions of the tax rates in period t and period t + 1. Formally, we obtain x c * t = X c (τ t , τ t+1 ) and (8) and (9) gives the comparative static results qq ) 2 > 0 due to stability reasons. An increase in the tax rate in a given period raises the marginal costs of food consumption and thereby reduces child and parent consumption in that period, as shown in (10). The fall in period t child consumption, following from an increase in the period t tax rate, in turn, has two opposing effects on parent consumption in t + 1: On the one hand, it reduces weight of the parent in t + 1, so the parent in t + 1 may increase its consumption during adulthood (due to γC qq > 0). On the other hand, the reduction in child consumption in t weakens habits in t + 1 and, thus, gives the parent in t + 1 an incentive to lower its own consumption (due to −V p xs < 0). Taking both effects together, the first expression in (11) shows that the impact of the period t tax rate on parent consumption in t + 1 is ambiguous. Similarly, the reduction in parent consumption in t + 1, following from an increase in the period t + 1 tax rate, reduces the long-term marginal costs perceived by the parent in t by lowering the marginal health costs, and increases the long-term marginal costs of stronger habits. Due to αγC qq > 0 and −αV p xs < 0 these changes in the perceived marginal costs translate into opposing effects on child consumption in t, so the parent in t may increase or decrease child consumption in t if the period t + 1 tax rate goes up, as shown by the second expression in (11).
3 In period 0, we obtain from (7) the additional condition V p x (x p * 0 , x c −1 )−1−τ 0 −C q (x p * 0 +γx c −1 ) = 0 where x c −1 is predetermined. This condition yields x p * 0 as a function of τ 0 . As we subsequently focus on the steady state only, we can safely ignore this condition from the initial period.

Optimal policy
The present value of welfare can be written as W = ∞ t=0 (u p t + u c t ). Inserting (1)-(4) and (6) as well as the public budget constraint t = τ t (x p * t + x c * t ) yields The optimal policy maximizes this welfare function, taking into account the comparative static effects (10) and (11). In determining the optimal fat tax rate in period t, we have to take into account the effects on period t child consumption x c * t = X c (τ t , τ t+1 ) and period t + 1 parent consumption x p * t+1 = X p (τ t , τ t+1 ). Moreover, the period t tax rate also influences period t − 1 child consumption x c * t−1 = X c (τ t−1 , τ t ) and period t parent consumption x p * t = X p (τ t−1 , τ t ). Differentiating (12) with respect to τ t and taking into account all these effects as well as (8) and (9) we obtain for t ∈ {1, 2, ...} 4 As Goulão and Pérez-Barahona (2014) we focus on the properties of the optimal tax in the steady state with τ t−1 = τ t = τ t+1 =: τ * . Inserting into (13) and solving gives Using this expression, we prove in Appendix B the following result.
Proposition. For any α ∈ [0, 1[ the optimal steady state fat tax rate τ * is strictly positive. In general, however, it deviates from the first best policy and is only second best. We obtain underinternalization (overinternalization) iff Increasing the fat tax reduces child consumption. This effect is intended since child consumption creates an externality. At the same time, the increase in the fat tax also 4 In t = 0, the first term in (13) vanishes, since x c −1 is predetermined. We can ignore this difference between t = 0 and all other periods, since we subsequently focus on the steady state only. changes parent consumption, which is not intended since parent consumption does not cause an externality. But this latter effect is of second order only, implying that the optimal fat tax rate is strictly positive, as stated in the first part of the proposition.
The unintended distortion of parent consumption explains why the optimal tax is not first best, as stated in the second part of the proposition. A tax rate increase in a given period reduces parent consumption in this period and, at first glance, one may conjecture that the optimal tax rate has to underinternalize the external marginal costs (1 − α)(γC q − V p s ) in order to mitigate the unintended reduction in parent consumption. But beside the intratemporal effect on parent consumption there is also an inter temporal effect on parent consumption in the next period which may lead to overinternalization: The intratemporal effect is reflected by ∂x p * t+1 /∂τ t+1 in (10), and ∂x p * t+1 /∂τ t in (11) gives the intertemporal effect, all expressions evaluated at the steady state. The intratemporal effect is negative, while the intertemporal effect is ambiguous; remember that it may be positive because the decline of consumption during childhood and the corresponding fall in weight in adulthood ceteris paribus induces the parent to eat more during adulthood. If the intertemporal effect is positive and larger in absolute terms than the intratemporal effect, then the fat tax has an unintended positive effect on steady state consumption of the parent and the optimal fat tax overinternalizes the external costs. In terms of the model primitives V p , V c and C, the conditions for under-and overinternalization are given in (15). An example with overinternalization is obtained if parents are non-altruistic (α = 0) and habits are absent (V p xs = V p ss = 0). For V c (x) = ax − bx 2 /2 and C(q) = cq 2 , overinternalization occurs if γc > b.

Conclusion
In this paper, we develop an OLG model to analyze non-altruism within the family as a rationale for fat taxes. We show that non-altruism is an argument for taxation of unhealthy food, indeed, but the optimal tax rate is only second best and may under-or overinternalize the intergenerational externality. Of course, this latter result relies on our implicit assumption that there is a uniform tax on parent and child consumption. If taxation may discriminate between parent and child consumption, a zero tax on the former and a tax equal to the marginal costs on the latter would do the job. However, in practice it is often difficult or even impossible to tax parents and children food consumption differently. Such discrimination may be possible if we take into account further margins. A thorough analysis of such margins is left for future research, though. (7) and (8) Iteratively inserting W t+1 in the objective function (5) yields

A Derivation of Equations
The parent in t maximizes (A.1) over x p t and x c t subject to (1)-(4) and (6), taking into account that it may affect its descendants' choices x j * t+i for j = p, c and i ≥ 1. It takes as given habits s t = x c t−1 already determined in t − 1. The first-order conditions are where we have replaced s t and q t by (2) and (3), respectively. If we iterate (A.2) and (A.3) to periods after t, we obtain the first-order conditions of the descendants' optimal choices and see that these first-order conditions do not depend on x p t . It follows Inserting this back into (A.2) proves (7). Moreover, using (7) Iterating (A.5) to periods after period t, we see that the resulting expression does not directly depend on x c t . They contain x p t+1+i for i ≥ 1. By the iterated version of (7), however, x p t+1+i only depends on x c t+i . Hence, the iterated version of (A.5) also does not depend indirectly on x c t (via x p t+1+i for i ≥ 1) and it follows

B Proof of the Proposition
Inserting the comparative static results (10) and (11) into Ω from (14) yields The properties of V p , V c and C as well as the conditions α ∈ [0, 1[ and γ ∈ [0, 1] imply Ω > 0 and, thus, a positive tax rate τ * > 0, as stated in the first part of the proposition. The first best policy is obtained by differentiating the welfare function directly with respect to x p t and x c t . Denoting the first best steady state values by x po and x co , respectively, the steady state first-order conditions read V p x (x po , x co ) − 1 − C q (x po + γx co ) = 0, (B.3) V c x (x co ) − 1 + V p s (x po , x co ) − γC q (x po + γx co ) = 0. (B.4) The steady state consumption levels x p * and x c * chosen by the parent satisfy (7) and (8). In the steady state, these conditions can be rewritten as If we insert τ * = (1 − α)(γC q − V p s )Ω from (14), we see that (B.5) does not coincide with (B.3). Hence, the optimal steady state tax rate is only second best. The first best requires a tax τ po = 0 on parent consumption and τ co = (1 − α)(γC q − V p s ) on child consumption. Under-(over-)internalization with respect to child consumption occurs if τ * < (>)(1 − α)(γC q − V p s ) or, equivalently, Ω < (>)1. Using (B.1) and rearranging gives (15) and completes the proof of the proposition.