Abstract
This paper studies the optimal linear tax-transfer policy in an economy where agents differ in productivity and in genetic background and where longevity depends on health spending and genes. If agents internalize imperfectly the impact of health spending on longevity, the utilitarian optimum can be decentralized with type-specific lump-sum transfers and Pigouvian taxes correcting for agents’ myopia and for their misperception of health spending’s effects on the economy’s resources. The second-best problem is examined under linear taxation instruments. It may be optimal to tax health spending, especially under complementarity of genes and health spending in the production of longevity.
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Notes
Sources: The Human Mortality Database (2009).
Note that longevity differentials are not here due to a selective involvement in wars: Swedish women born in 1900 did not experience any world wars.
Note that, in the presence of addiction, whether mortality factors like obesity, alcoholism, and drug consumption are under the control of agents or not is questionable.
Data: World Health Organization Statistical Information System (2009). The sample includes 193 countries.
Environmental factors of longevity include the quality of lands (Kjellström 1986), of waters (Sartor and Rodia 1983), and of the air (Kinney and Ozkanyak 1991). Genetic diseases take various forms, such as the sickle-cell disease and the familial hypercholesterolemy (favoring heart attacks; Soliani and Lucchetti 2001).
That picture comes from the study of Herskind et al. (1996), which relies on a sample of 2,872 Danish twin pairs born between 1870 and 1900.
That assumption is supported by the large empirical literature on limited rationality in the context of health-affecting behavior (see O’Donoghe and Rabbin 2000). In particular, the widespread feeling of invulnerability of young people shown by Quadrel et al. (1993) can be regarded as some myopia, i.e., an ignorance of the effects of one’s actions.
Note that we are not dealing here with the idea that individuals might rationally adopt a high rate of discount. In that case, government’s intervention is highly questionable and has been labeled “old paternalism”.
Thus, the critical utility level for continuing existence is set to zero (see Broome 2004).
Our focus on longevity-enhancing spending has important consequences when interpreting the results of this study. In reality, various health spending, which have little relationship with longevity, exhibit a strong redistributive dimension, or affect the quality of life periods or productivity. Whether such spending should be subsidized lies outside the scope of this paper, which focuses on longevity-improving spending.
Thus, this study complements other papers, such as Zhang et al. (2006) and Pestieau et al. (2008), which analyze the optimal taxation policy in a dynamic framework, but without an explicit heterogeneity in longevity-enhancing characteristics. An exception is Ponthiere (2010), who studies lifestyle-based longevity in a dynamic model where lifestyles are transmitted vertically or obliqually across generations.
This expression presupposes no pure time preferences, as well as a utility from being dead normalized to zero.
Note that this formalization of myopia is formally equivalent to assuming some pessimism of agents, in the sense that, under α < 1, the perceived probability of survival is always inferior to the actual probability. While this constitutes a simplification, that modeling of myopia has the virtue of analytical conveniency.
We will discuss the implications of those assumptions throughout this paper.
The assumption of a perfect annuity market, which is rather strong, is made here for analytical convenience. Actually, this study would like to abstract from tractability difficulties raised by accidental bequests, which are examined by Cremer et al. (2007). Assuming a perfect annuity market for each risk class is one way to avoid those difficulties. Another way consists in assuming that the government taxes entirely the savings of the dead (see Section 4).
A concern for responsibility implies that one pays attention, to some extent, to the relation between the initial conditions in which agents are and their final positions. Here, the social planner has, as a unique objective, the maximization of the sum of utilities (which, under welfarism, are the unique relevant pieces of information for positions). That maximization problem is only constrained by survival functions and utility functions and, thus, does not pay a specific attention to how conditions and positions are related.
The difficulties raised by varying longevity include, among other things, the definition of a critical utility level for continuing existence, making the addition of a new life-period yielding that utility level neutral. In the following, we rely on classical utilitarianism, where that critical level is fixed to zero.
Note also that if α i = α < 1 ∀ i, it would follow that θ i = θ ∀ i.
Equivalently, we assume that R i = 1 ∀ i. The absence of an annuity market consists of another major second best feature of the framework studied in this section.
A rate below 100% would complicate the analysis significantly, as some agents would benefit from unintended bequests. This would be, to some extent, close to the assumption of imperfect annuity markets, discussed in Cremer et al. (2007). For simplicity, we shall abstract here from those difficulties by assuming full taxation of the saving of dead agents.
It is also assumed, for simplicity, that agents cannot borrow with public pensions as collateral.
Here again, we assume, for simplicity, that an interior solution exists. See Andersen and Bhattacharya (2008) on the savings of myopic agents under a PAYG system.
To simplify the presentation, we shall assume, in this section that θ = P = 0, so that changes in τ affect the demogrant T only.
Total differentiation of the aggregate households’ budget constraint \(\frac{\left( 1-\tau \right) ^{2}Ew^{2}}{2}+T=0\) leads to \(\left( 1-\tau \right) Ew^{2}d\tau =dT\).
On that technique, see Cremer et al. (2008). There are other ways to combine the first-order conditions. One way allows to get rid of the multiplier μ, but the expression obtained by following that alternative approach would have a more difficult interpretation.
For conveniency, we shall here delete the * superscripts, but we remain at the optimum.
We now assume that θ = T = 0.
Note that here again, these are not real substitution effects, as we consider aggregate compensation.
For the ease of presentation, we now assume here that τ = P = 0.
The term \(\bar{e}\) denotes the average level of health spending in the population.
Take the case of two agents 1 and 2 with w 2 > w 1 > 0 and ε 2 > ε 1 = 0, and assume \(\pi \left( \varepsilon _{i}+e_{i}\right)\). Assume further that \(\pi _{1}\left( 0\right) =0\), \(\pi _{1}^{\prime}\left( 0\right) =\infty ,\pi _{2}(\varepsilon _{2})=1,\) \(\pi _{2}^{\prime}(\varepsilon _{2})=0\). One expects e 1 > 0 and e 2 = 0.
If e and ε are substitutable, \(cov\left( u^{\prime }\left( x\right) ,e\right) >0\), and we have a positive or a negative tax on health.
Clearly, if only e can enhance longevity, more productive agents do not spend less on health than less productive agent, as there exists, under π ε = 0, no way to ‘compensate’ low health spending in longevity terms.
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Acknowledgements
The authors are most grateful to Dirk Van de gaer for his comments and suggestions on this paper. We would like also to thank two anonymous referees for their helpful remarks on this paper.
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Leroux, ML., Pestieau, P. & Ponthiere, G. Optimal linear taxation under endogenous longevity. J Popul Econ 24, 213–237 (2011). https://doi.org/10.1007/s00148-010-0304-1
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DOI: https://doi.org/10.1007/s00148-010-0304-1