---
title: "The Taxation of Life Annuities Under Adverse Selection"
authors:
  - name: "Alexis Direr"
    affiliation: "Paris School of Economics; INRA; Université de Grenoble 2"
date: "2009-04"
doi: "10.1016/j.jpubeco.2009.11.006"
keywords: [optimal taxation, asymmetric information, life-cycle models, saving, life annuities, adverse selection, longevity, Mirrlees]
jel_codes: [H21, D82, D91]
language: en
type: research-article
journal: "Journal of Public Economics"
---

# The Taxation of Life Annuities Under Adverse Selection

**Author**
- Alexis Direr — Paris School of Economics; INRA; Université de Grenoble 2 — *direr@ens.fr*

**DOI**: [10.1016/j.jpubeco.2009.11.006](https://doi.org/10.1016/j.jpubeco.2009.11.006)

**Keywords**: optimal taxation, asymmetric information, life-cycle models, saving, life annuities, adverse selection.
**JEL codes**: H21, D82, D91.

---

## Abstract

This paper studies how annuities should be taxed in a model à la Mirrlees (1971) in presence of adverse selection and a positive link between income and longevity. An annuity tax can address the adverse selection problem by subsidizing small annuities (purchased by low income groups) and taxing large annuities (purchased by high income groups). Numerical simulations suggest that the degree of progressivity of taxation is significant and increases when annuitants get older.

---

## 1. Motivation and contributions

Most developed countries promote private retirement saving by exempting from income tax either contributions during accumulation or benefits during the payout phase (Whitehouse 1999; Antolin et al. 2004). This favourable treatment is hard to justify on standard public-finance grounds: under the Atkinson–Stiglitz (1976) theorem, a non-linear income tax combined with weakly separable utility makes commodity taxation — and by extension saving taxation — redundant. The paper asks whether life annuities are an exception to this benchmark.

The paper makes the following contributions.

1. **Two distinct rationales for taxing annuities under adverse selection and longevity gradients.** In a Mirrlees (1971) model with a continuum of skills, one working period and many retirement periods, the paper isolates two motives that justify a non-trivial annuity tax. **(i) An actuarial-correction motive**: insurers cannot observe individual mortality and offer a pooling rate of return, which underprices annuities for the long-lived (rich) and overprices them for the short-lived (poor); a tax can restore individual actuarial fairness. **(ii) A redistributive luxury-good motive**: because the rich live longer, they consume more retirement annuities in expectation, so taxing annuities relaxes the redistributive constraints on the income tax.

2. **Analytical characterization of the optimum and ambiguity of progressivity in closed form.** The two motives push in opposite directions on the slope of the annuity tax: the actuarial motive alone implies progressivity, while the luxury-good motive — under the standard finding that $T'(wL)$ declines over most of the income distribution (Seade 1977; Diamond 1998) — generates regressivity. The model alone cannot sign the overall slope.

3. **Generalization of Brunner and Pech (2008).** Their two-type result (annuities of the most productive should be taxed) is shown to be a *local* property at the upper bound of the skill distribution, where the marginal income tax is zero. The continuous-skill, multi-period framework here delivers the full schedule and its evolution with age.

4. **Quantitative calibration on French data.** Simulations using the Bourguignon–Spadaro (2007) wage distribution and the Bommier et al. (2005) survival-rate elasticities show that the actuarial-correction motive dominates quantitatively, generating a progressive schedule that subsidizes low annuities and taxes high annuities. Marginal tax rates on annuities reach up to 40% at age 90 for the richest, and progressivity steepens with age. Welfare gains are sizable: introducing the optimal annuity tax is equivalent to raising consumption of the poorest by 5% per period and lowering consumption of the richest by 6% per period.

---

## 2. Model

**Demographics and preferences.** Continuum of agents with productivity $w \in W = [\underline{w}, \overline{w})$, distribution $F$, density $f$. One working period (date 1) followed by $n-1$ retirement periods. Survival probability at age $i$ conditional on being alive at date 1 is $\pi_i(w) > 0$, with $\pi_1(w) = 1$ and $\pi'_i(w) \geq 0$ (longevity rises with skill). Time-separable lifetime utility:

$$U(C, L, w) = \sum_{i=1}^{n} \beta^{i-1}\pi_i(w)\, u(c_i) + v(L)$$

with $u$ concave, $v$ convex, $U_{Li}=0$ but $U_{wi}>0$ for $i \geq 2$ — utility is non-separable between consumption and skill *because* skill enters the survival probability.

**Annuity market with pooling.** Following Yaari (1965) and Abel (1986), agents fully annuitize. Insurers cannot observe individual mortality and cannot price-discriminate on quantity (savers can split wealth across insurers). Zero-profit per period implies the pooling rate of return:

$$Q_i = \frac{\int_W (c_i + t_i(c_i))\, dF(w)}{\int_W (c_i + t_i(c_i))\,\pi_i / R^{i-1}\, dF(w)}$$

where $R$ is one plus the safe rate. $Q_i$ is a $\pi_i$-weighted average of population survival rates with annuity-demand weights.

**Tax instruments.** The government uses a non-linear income tax $T(wL)$ and age-specific non-linear annuity taxes $t_i(c_i)$, $i \geq 2$, with $t_1 \equiv 0$ (a uniform first-period commodity tax is absorbed into $T$). It maximizes $\int \Psi(U(w))\, dF(w)$ subject to the consolidated budget:

$$\int_W \Big( wL - \sum_{i=1}^{n} \pi_i c_i / R^{i-1} \Big) dF(w) = PS$$

and to incentive compatibility (Mirrlees first-order approach).

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## 3. Analytical results

**Optimal income tax (Eq. 4).** Taking the same form as in static Mirrleesian models:

$$\frac{T'(wL)}{1 - T'(wL)} = A(w)\, B(w)\, D(w)$$

with $A(w) = 1 + 1/\varepsilon(w)$ (labor supply elasticity), $B(w) = (1 - F(w))/(wf(w))$ (distributional weight) and $D(w)$ summarizing the social value of redistribution. The intertemporal structure of the problem leaves the income-tax formula formally unchanged (Diamond 2003).

**Optimal annuity tax (Eq. 5).** For $i > 1$:

$$1 + t'_i(c_i(w)) = H_i(w)\, \big(1 - \sigma_i(w)\, B(w)\, D(w)\big)^{-1}$$

with

$$H_i(w) = \frac{\pi_i(w)\, Q_i}{R^{i-1}}, \qquad \sigma_i(w) = \frac{w\, \pi'_i(w)}{\pi_i(w)}$$

The two factors map directly onto the two motives.

### 3.1 Actuarial-correction term $H_i(w)$

When inserted into the Euler equation, $H_i$ exactly offsets the survival-rate dependence in the actuarial rate of return $\pi_i(w) Q_i$, restoring the symmetric-information intertemporal marginal rate of transformation $R^{i-1}$.

> **Lemma 1.** Define the threshold skill $\widetilde{w} \in (\underline{w}, \overline{w})$ by $\pi_i(\widetilde{w}) = \int_W (c_i + t_i(c_i))\pi_i\, dF / \int_W (c_i + t_i(c_i))\, dF$. With only the $H_i$ term active, $t'_i(c_i(w)) \geq 0$ for $w \geq \widetilde{w}$ and $t'_i(c_i(w)) < 0$ for $w < \widetilde{w}$.

The actuarial motive alone subsidizes the poorest annuitants and taxes the richest. When $\pi_i(w) = \pi_i$ (homogeneous mortality), $\sigma_i = 0$, $Q_i = R^{i-1}/\pi_i$, and the annuity tax collapses to zero — recovering Atkinson–Stiglitz (1976).

### 3.2 Luxury-good term $(1 - \sigma_i B D)^{-1}$

The factor reflects the redistributive premium attached to retirement consumption: because $U_{wi} > 0$, retirement consumption is more valuable to the more productive. Higher elasticity of survival to skill $\sigma_i$ raises the optimal annuity tax rate ceteris paribus.

### 3.3 Slope of the annuity tax: ambiguity in closed form

Eq. (6) re-expresses the schedule in terms of the income marginal rate:

$$1 + t'_i(c_i(w)) = \frac{\pi_i(w) Q_i}{R^{i-1}} \left(1 - \frac{\sigma_i(w)}{1 + 1/\varepsilon(w)}\, \frac{T'(wL)}{1 - T'(wL)}\right)^{-1}$$

The first factor rises with skill (actuarial motive: progressive). The second factor moves with $T'(wL)$, which in standard Mirrleesian solutions is decreasing across most of the distribution (regressive luxury-good effect). The overall slope is therefore not pinned down analytically.

A second-order caveat: the multiplicative factor can in principle approach unity, generating an interior kink in the budget constraint and a gap in the consumption distribution. The simulations check ex post that earnings and consumption rise monotonically in skill, satisfying Mirrlees (1971) second-order conditions.

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## 4. Calibration

**Demographic and preference parameters.**

| Parameter | Value | Source / rationale |
|---|---|---|
| Maximum age | 100 | — |
| Period utility | $u(c) = c^{1-\sigma}/(1-\sigma)$ | CRRA |
| Risk aversion $\sigma$ | 2 | $1/\sigma = 0.5$, standard macro calibration |
| Labor disutility | $v(L) = (\varepsilon/(1+\varepsilon))\beta L^{1+1/\varepsilon}$ | exponential |
| Labor elasticity $\varepsilon$ | 0.3 | French estimates: 0.1–0.2 men, 0.5 married women (Bourguignon–Spadaro 2007) |
| Riskless rate $R-1$ | 4% | — |
| Discount rate | $\beta = 1/R$ | flat consumption profile in first best |
| Government spending | $PS / Y = 0.30$ | — |
| Welfare criterion | $\Psi'=1$ | Utilitarian |

**Wage distribution.** Bourguignon and Spadaro (2007) French wage distribution for 1995, kernel-smoothed; right tail truncated at $\overline{w} = 10\,\underline{w}$; 10% non-workers atom at the bottom. Average income normalized to unity in figures.

**Survival-rate elasticities to skill $\sigma_i(w)$.** Sourced from Bommier et al. (2005), who estimate how survival rates vary with French public-pension benefits (a near-proportional proxy for wages). Values range from **0.019** at age 67 to **0.043** at age 91 for men, and are much smaller for women (Figure 2 of the paper). The simulation uses a sex-averaged interpolation.

**Wage-dependent mortality table.** Constructed in two steps. (i) The mortality table of male unskilled workers (Robert-Bobée and Monteil 2005, French deaths in mid-1990s) is assigned to the lowest skill $\underline{w}$. (ii) Instantaneous survival probabilities at age $i$, $p_i(w) = \pi_i(w)/\pi_{i-1}(w)$, evolve along the wage distribution as $p_i = k\, w^{\sigma_i}$, with $k$ pinned down so that $p_i(\underline{w})$ matches step (i). The implied life-expectancy gap at age 65 between $\underline{w}$ and $\overline{w}$ is **6 years** (80.1 vs. 86.4 years), consistent in scale with the 4-year gap Robert-Bobée–Monteil document between French male unskilled workers and male executives.

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## 5. Numerical results

### 5.1 Optimal income tax (Figure 3)

Marginal income tax rate is zero at the top (bounded skill distribution), strictly positive at the bottom (Seade 1977, due to non-workers), and decreases monotonically over most of the income range — consistent with Mirrlees (1971) and Tuomala (1990).

### 5.2 Marginal annuity tax rates by age and annuity size (Figures 4–6)

The schedule is **progressive at every age**, starting from negative values for small annuities and rising with annuity size:

- **Age 70**: range from roughly $-0.05$ to $+0.05$ across the annuity distribution.
- **Age 80**: range roughly $-0.20$ to $+0.20$.
- **Age 90**: range roughly $-0.40$ to $+0.50$.

Plotted by age (Figure 6), marginal tax rates fan out: low-wage individuals see their rates of return *subsidized* (negative marginal rate), while high-wage individuals face marginal rates rising with age, reaching about **40%** at age 90 for the highest skill.

> **Authors' critical reading.** Comparison of the optimal schedule to the "actuarially fair" tax — the schedule that would emerge from the actuarial-correction motive alone, $1 + t'_i = \pi_i(w) Q_i / R^{i-1}$ — is visually almost indistinguishable from the full optimum (Figure 5). The luxury-good motive contributes only marginally to the quantitative shape. The system is essentially designed to neutralize rate-of-return inequalities arising from longevity heterogeneity. A *linear* annuity tax could not perform this job.

### 5.3 Average tax rate on annuities by wage and age (Figure 7)

Normalizing $t_i(c_i(\underline{w})) = 0$ for every age (an innocuous normalization given that only the intertemporal sum of taxes affects the consumer's budget), the **U-shape** of the average rate $t(c_i)/(c_i + t(c_i))$ across wages becomes visible:

- Subsidies up to roughly 5% are concentrated around the **median wage** (maximum subsidy at the median).
- Wages above twice the median are taxed, with the rate rising in wage and steepening with age (reaching roughly 12% at $3 \times$ median wage at age 90).

### 5.4 Welfare gains (Figure 8)

Defining $\delta(w)$ as the uniform consumption rate of variation that equates lifetime utility in the optimum-annuity-tax economy and the no-annuity-tax economy (income tax re-optimized with the same $PS/Y$):

$$U((1+\delta(w))\,C^*, L^*, w) = U(\widehat{C}, \widehat{L}, w)$$

implementing the annuity tax is equivalent to raising consumption of the poorest by **+5%** at every age and reducing consumption of the richest by **−6%**. The welfare scope of the redistribution is thus economically substantial.

### 5.5 Sensitivity analysis (Appendix 5)

Eight summary indicators are computed for the lowest wage quintile (Q1) and the upper decile (D10): average total-tax-burden ratio (A), average annuity tax rate at age 80 (B), average marginal annuity tax rate at age 80 (C), and the welfare-equivalent consumption variation (D). Selected results from the paper's Table:

| Indicator | Baseline | $\varepsilon=0.25$ | $\varepsilon=0.35$ | $\beta=0.95$ | $\beta=0.97$ | $\sigma=1$ | $\sigma=3$ |
|---|---:|---:|---:|---:|---:|---:|---:|
| $C_{Q1}$ — avg. marg. annuity tax, Q1 (%) | −18.25 | −18.23 | −18.28 | −18.25 | −18.25 | −19.08 | −16.86 |
| $C_{D10}$ — avg. marg. annuity tax, D10 (%) | 15.33 | 15.33 | 15.33 | 15.33 | 15.33 | 15.20 | 15.44 |
| $D_{Q1}$ — welfare-equivalent $\delta$, Q1 (%) | 4.40 | 4.39 | 4.41 | 4.24 | 4.55 | 9.46 | 2.61 |
| $D_{D10}$ — welfare-equivalent $\delta$, D10 (%) | −4.14 | −4.00 | −4.00 | −3.88 | −4.14 | −7.09 | −2.66 |

Results are quantitatively stable to $\varepsilon$ and $\beta$ and somewhat more sensitive to $\sigma$. Welfare gains rise in the intertemporal elasticity of substitution $1/\sigma$: when consumption smoothing is easier, the low-skilled distort their consumption profile more aggressively to absorb the lower actuarial return implied by adverse selection, magnifying the value of correcting it.

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## 6. Conclusion

Most OECD tax systems exempt either annuity contributions or annuity payouts. The model here implies the opposite policy at the margin: **only the smallest annuities should be subsidized, while the largest should be taxed**, with progressivity steepening with age. The dominant mechanism is the correction of actuarially-unfair pooling that disadvantages short-lived (poor) annuitants and advantages long-lived (rich) annuitants. The redistributive luxury-good motive plays a secondary quantitative role.

### Limitations and research extensions

The author flags four extensions left for future work.

1. **Separating equilibria.** If insurers can offer different time-payout profiles (Brunner and Pech 2005) or price-quantity bundles (Eckstein, Eichenbaum and Peled 1985), self-selection could partially undo adverse selection. Empirically Finkelstein and Poterba (2004) document such self-selection in the UK annuity market, but their companion paper (2002) supports the quantitative importance of residual adverse selection. Were adverse selection assumed away, only the luxury-good motive would remain.
2. **Bond holding.** The model assumes annuities dominate bonds. Because the actuarial-correction motive flattens after-tax expected returns to annuities just enough to leave a mortality premium over bonds, the assumption appears robust.
3. **Stochastic income–longevity link.** A deterministic mapping between skill and survival probabilities is assumed. Replacing it with a noisy correlation should attenuate progressivity.
4. **Annuity puzzle.** Empirically, individuals annuitize less than the model predicts (Davidoff, Brown and Diamond 2005), for reasons including bequest motives, medical-expenditure risk, public pensions, and behavioral factors (Brown 2007). Embedding these features would likely lower the overall level of optimal annuity taxation.

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## Acknowledgments

The author thanks Amadeo Spadaro for providing the French wage data. The paper benefited from comments by Johann Brunner, Gwenola Trotin, and Thomas Weitzenblum, and from feedback received at the international workshop on Longevity and Annuitization (Paris, June 2008), the 7th Journées Louis-André Gérard-Varet (Marseille, June 2008), and the annual meeting of the Public Economic Theory association (Seoul, June 2008).

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## Main references

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Atkinson, A. B., Stiglitz, J. E. (1976). The Design of Tax Structure: Direct versus Indirect Taxation. *Journal of Public Economics* 6, 55–75.

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*The full reference list appears in the PDF.*