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On the optimal lifetime redistribution and social objectives: a multidimensional approach

Abstract

We characterize optimal redistribution policy when there are differences not only in individuals’ productivities but also in their tastes towards the timing of consumption, i.e. some are patient and others impatient in consumption over the life cycle and this preference together with productivity is non-observable to government. We consider different social objectives and incorporate a novel approach taken in the spirit of Roemer (Equality of opportunity, Harvard University Press, Harvard, 1998) and Van de Gaer (Equality of opportunity and investments in human capital, Katholieke Universiteit Leuven, 1993). This approach applies a compromise between the principle of compensation and the principle of responsibility. We derive analytical expressions which describe the optimal distortion (upward or downward) in saving. As the multidimensional problems become very complicated, to gain a better understanding, we also numerically examine the properties of an optimal lifetime redistribution policy. We find support for a nonlinear tax/pension program in which impatient types are taxed at the margin, and patient low ability types are subsidized in their retirement consumption. Numerical simulations show quite big differences in terms of the levels of marginal tax rates between different social objectives, indicating that the optimal income taxation results are sensitive to the choice of the social planner’s goals.

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Fig. 1

Notes

  1. One interpretation for the differences in discount factors beside consumption preferences could be that there are individuals who expect to live shorter lives and therefore emphasize the first period consumption. Fleurbaey et al. (2014) study these kind of longevity differences and redistribution.

  2. Bossert (1995) and Fleurbaey (1994) have studied the idea of compensating inequalities due to circumstances only, while leaving other inequalities untouched.

  3. Fleurbaey and Maniquet (2006) advocate a social welfare function based on fairness principles that puts a greater weight to “working poors” if preferences differ towards leisure. In an intertemporal model like ours this could mean that the more patient poor should have a greater weight. However, as discussed, we are more agnostic about this normative dimension and instead use the Roemer social welfare function as described in the next paragraph.

  4. Tenhunen and Tuomala (2010) studied optimal lifetime redistribution in a 4-types setting where government’s objective is either utilitarian or paternalistic and consumer preferences are approximated with Cobb–Douglas utility function. Tenhunen and Tuomala (2013) studied how habit formation affects the optimal tax and pension scheme under heterogeneous preferences.

  5. Sandmo (1993) considers a case where people differ in preferences, but are endowed with the same resources. Tarkiainen and Tuomala (1999, 2007) also consider a continuum of taxpayers simultaneously distributed by skill and preferences for leisure and income.

  6. Alternatively the same outcome could be reached by assuming homothetic preferences and linear Engel curves.

  7. Another specification for utility function could be \(U^{i}=u(c^{i})/\delta ^{i}+v(x^{i})+\psi (1-y^{i})\) which would imply that the ones with higher savings rate are less willing to increase work for additional money. We justify our choice for the utility representation as we are considering lifetime redistribution where the timing of the retirement consumption is a far-off event instead of a nearby event, and in this kind of setting the representation in the text is standard. We also want to compare our results to earlier studies, which are done with this specification. It is obvious that a different choice of utility representation will engender a different optimal solution and affect our results. Effectively the problem is reversed and different IC constraints are binding than in the current setting. See also discussion in Diamond and Spinnewijn (2011).

  8. The direction of the binding self-selection constraint is assumed to be, following the tradition in the one-dimensional two-type model, from high-skilled individual towards low-skilled individual. This pattern is also confirmed in the numerical simulations.

  9. In the numerical solution we also consider the marginal labour income tax rates. As has become conventional in the literature we may interpret the marginal rate of substitution between gross and net income as one minus the marginal income tax, \(\frac{\psi ^{'}(\frac{ny}{n})}{nu_{c}}=1-T'(ny)\), which would be equivalent to the characterization of the labour supply of an agent facing an income tax function \(T'(ny)\). As in our model the heterogeneity shows up in the discount factor of the second period instant utility, the analytical results do not differ for the optimal labour income tax for the two distinct preference groups. The marginal labour income tax rates satisfy the usual properties; \(T(n^\mathrm{L}y^\mathrm{L})>0\) and \(T(n^\mathrm{H}y^\mathrm{H})=0\).

  10. The numerical procedure is described in Tenhunen and Tuomala (2010) “Appendix B”.

  11. The slackness of the other self-selection constraints is also checked by calculating the difference in utilities when mimicking and when not.

  12. In the case of perfect negative correlation, numerical simulations show that L’s distortion is a marginal subsidy. See Fig. 1 in Sect. 4 for results with varying correlation.

  13. In the case of one-dimensional heterogeneity types are ordered usually with respect to their income, consumption or utilities but in a two-dimensional world the ordering is not self-evidently clear.

  14. Due to solvability problems the results with CES utility functions are given with parametric values \(N^{1}=0.5, N^{3}=0.254\) and \(N^{4}=0.246\).

  15. In fact, with the CES function, in order to see what kind of effects the discount rates have for the results, the size of the groups needs to be modified: the sizes of the groups in this exercise are set to \(N^{1}=0.2, N^{2}=0.3\) and \(N^{3}=0.5\).

  16. Or we can first calculate the average utility in each skill group and then apply the maximin criterion to such average figures.

  17. For example there is no binding constraint in the zero correlation case.

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Acknowledgements

We would like to thank two anonymous referees for valuable comments. We also thank the seminar audiences at the 34th Summer Seminar of Finnish Economists, the XXXVII Annual Meeting of the Finnish Economic Association, the 71st Annual Congress of the International Institute of Public Finance in Dublin and the Nordic workshop on tax policy and public economics in Oslo for the discussion.

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Correspondence to Terhi Ravaska.

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Funding from the Strategic Research Council of Academy of Finland, No. 293120 (STN-WIP-consortium) is gratefully acknowledged. Ravaska also gratefully acknowledges the funding from OP Groups Scientific Foundations.

Appendices

Appendix A: First-order conditions

To shorten the notation, we denote the partial derivatives as follows: \(\frac{du(c^{i})}{dc^{i}}=u_{c}^{i}, \frac{\mathrm{d}v(x^{i})}{\mathrm{d}x^{i}}=v_{x}^{i}\) and \(\frac{\mathrm{d}\psi (1-y^{i})}{\mathrm{d}y^{i}}=\psi ^{'i}\)

Two types

The Lagrange of the optimization problem is

$$\begin{aligned} \begin{aligned} \mathcal {L}&= N^\mathrm{L}[u(c^\mathrm{L})+ \delta ^\mathrm{L}v(x^\mathrm{L})+\psi (1-y^\mathrm{L})]\\&\quad + \lambda \left[ \sum N^{i}(n^{i}y^{i}-c^{i}-rx^{i})-R\right] \\&\quad +\mu ^\mathrm{HL} \left[ u(c^\mathrm{H})+\delta ^\mathrm{H}v(x^\mathrm{H})+\psi (1-y^\mathrm{H})\right. \\&\left. \quad -u(c^\mathrm{L})-\delta ^\mathrm{H}v(x^\mathrm{L})-\psi \left( 1-\frac{n^\mathrm{L}}{n^\mathrm{H}}y^\mathrm{L}\right) \right] . \end{aligned} \end{aligned}$$
(A.1)

The first-order conditions with respect to \(c^{i}, x^{i}\) and \(y^{i}, i=L,H\) are

$$\begin{aligned}&N^\mathrm{L}u_{c}^\mathrm{L} - \lambda N^\mathrm{L} - \mu ^\mathrm{HL}u_{c}^\mathrm{L} = 0 \end{aligned}$$
(A.2)
$$\begin{aligned}&N^\mathrm{L} \delta ^\mathrm{L} v_{x}^\mathrm{L} - \lambda r N^\mathrm{L} - \mu ^\mathrm{HL}\delta ^\mathrm{H}v_{x}^\mathrm{L} = 0 \end{aligned}$$
(A.3)
$$\begin{aligned}&-N^\mathrm{L}\psi ^{'L} + \lambda N^\mathrm{L} n^\mathrm{L} + \mu ^\mathrm{HL}\frac{n^\mathrm{L}}{n^\mathrm{H}}\psi ^{'L} = 0 \end{aligned}$$
(A.4)
$$\begin{aligned}&-\lambda N^\mathrm{H} + \mu ^\mathrm{HL}u_{c}^\mathrm{H} = 0 \end{aligned}$$
(A.5)
$$\begin{aligned}&-\lambda r N^\mathrm{H} + \mu ^\mathrm{HL}\delta ^\mathrm{H}v_{x}^\mathrm{H} = 0 \end{aligned}$$
(A.6)
$$\begin{aligned}&\lambda N^\mathrm{H} n^\mathrm{H} - \mu ^\mathrm{HL}\psi ^{'H} = 0 \end{aligned}$$
(A.7)

Three types: low-productivity workers pooled

Using the information of the binding self-selection constraints provided by numerical solution, the Lagrange function in the case of maximin objective function can be written as

$$\begin{aligned} \mathcal {L}= & {} N^{1}[u(c^\mathrm{L})+\delta ^\mathrm{L}v(x^\mathrm{L})+\psi (1-y^\mathrm{L})]\nonumber \\&+\,\lambda \left[ \sum N^{i}(n^{i}y^{i}-c^{i}-rx^{i})-R\right] \nonumber \\&+\,\mu ^{43}[u(c^{4})+\delta ^\mathrm{H}v(x^{4})+\psi (1-y^{4})\nonumber \\&-\, {u}(c^{3})-\delta ^\mathrm{H}v(x^{3})-\psi (1-y^{3})] \nonumber \\&+\,\mu ^{31}\left[ u(c^{3})+\delta ^\mathrm{L}v(x^{3})+\psi (1-y^{3})- {u}(c^{1})\right. \nonumber \\&-\left. \delta ^\mathrm{L}v(x^{1})-\psi \left( 1-\frac{n^\mathrm{L}}{n^\mathrm{H}}y^{1}\right) \right] \end{aligned}$$
(A.8)

The first-order conditions with respect to \(c^{i}, x^{i}\) and \(y^{i}, i=1,3,4\) are given by

$$\begin{aligned} N^{1}u_{c}^{1} - \lambda N^{1} - \mu ^{31}u_{c}^{1}&= 0 \end{aligned}$$
(A.9)
$$\begin{aligned} N^{1} \delta ^\mathrm{L} v_{x}^{1} - \lambda r N^{1} - \mu ^{31}\delta ^\mathrm{L} {v}_{x}^{31}&= 0 \end{aligned}$$
(A.10)
$$\begin{aligned} N^{1}\psi ^{'} - \lambda N^{1} n^\mathrm{L} - \mu ^{31}\frac{n^\mathrm{L}}{n^\mathrm{H}} {\psi }^{'31}&= 0 \end{aligned}$$
(A.11)
$$\begin{aligned} - \lambda N^{3} - \mu ^{43} {u}_{c}^{43}+\mu ^{31}u_{c}^{3}&= 0 \end{aligned}$$
(A.12)
$$\begin{aligned} - \lambda r N^{3} - \mu ^{43}\delta ^\mathrm{H} {v}_{x}^{43}+\mu ^{31}\delta ^\mathrm{L}v_{x}^{3}&= 0 \end{aligned}$$
(A.13)
$$\begin{aligned} -\lambda N^{3} n^\mathrm{H} - \mu ^{43} {\psi }^{'43}+\mu ^{31} {\psi }^{'31}&= 0 \end{aligned}$$
(A.14)
$$\begin{aligned} -\lambda N^{4} + \mu ^{43}u_{c}^{4}&= 0 \end{aligned}$$
(A.15)
$$\begin{aligned} -\lambda r N^{4} + \mu ^{43}\delta ^\mathrm{H}v_{x}^{4}&= 0 \end{aligned}$$
(A.16)
$$\begin{aligned} -\lambda N^{4} n^\mathrm{H} + \mu ^{43}\psi ^{'}&= 0 \end{aligned}$$
(A.17)

In the utilitarian case, the Lagrange function with binding incentive–compatibility constraints can be written as

$$\begin{aligned} \mathcal {L}= & {} \sum N^{i}[u(c^{i})+\delta ^{i}v(x^{i})+\psi (1-y^{i})] \nonumber \\&+\,\lambda \left[ \sum N^{i}(n^{i}y^{i}-c^{i}-rx^{i})-R\right] \nonumber \\&+\,\mu ^{43}[u(c^{4})+\delta ^\mathrm{H}v(x^{4})+\psi (1-y^{4})\nonumber \\&-\, {u}(c^{3})-\delta ^\mathrm{H}v(x^{3})-\psi (1-y^{3})] \nonumber \\&+\,\mu ^{31}\left[ u(c^{3})+\delta ^\mathrm{L}v(x^{3})+\psi (1-y^{3})\right. \nonumber \\&-\left. {u}(c^{1})-\delta ^\mathrm{L}v(x^{1})-\psi \left( 1-\frac{n^\mathrm{L}}{n^\mathrm{H}}y^{1}\right) \right] . \end{aligned}$$
(A.18)

The first-order condition with respect to \(c^{i}, x^{i}\) and \(y^{i}, i=1,3,4\) are given by

$$\begin{aligned} N^{1}u_{c}^{1} - \lambda N^{1} - \mu ^{31}u_{c}^{1}&= 0 \end{aligned}$$
(A.19)
$$\begin{aligned} N^{1} \delta ^\mathrm{L} v_{x}^{1} - \lambda r N^{1} - \mu ^{31}\delta ^\mathrm{L} {v}_{x}^{31}&= 0 \end{aligned}$$
(A.20)
$$\begin{aligned} N^{1}\psi ^{'1} - \lambda N^{1} n^\mathrm{L} - \mu ^{31}\frac{n^\mathrm{L}}{n^\mathrm{H}} {\psi }^{'31}&= 0 \end{aligned}$$
(A.21)
$$\begin{aligned} N^{3}u_{c}^{3} - \lambda N^{3} - \mu ^{43} {u}_{c}^{43}+\mu ^{31}u_{c}^{3}&= 0 \end{aligned}$$
(A.22)
$$\begin{aligned} N^{3} \delta ^\mathrm{L} v_{x}^{3}- \lambda r N^{3} - \mu ^{43}\delta ^\mathrm{H} {v}_{x}^{43}+\mu ^{31}\delta ^\mathrm{L}v_{x}^{3}&= 0 \end{aligned}$$
(A.23)
$$\begin{aligned} N^{3}\psi ^{'3}-\lambda N^{3} n^\mathrm{H} - \mu ^{43} {\psi }^{'43}+\mu ^{31} {\psi }^{'31}&= 0 \end{aligned}$$
(A.24)
$$\begin{aligned} N^{4}u_{c}^{4}-\lambda N^{4} + \mu ^{43}u_{c}^{4}&= 0 \end{aligned}$$
(A.25)
$$\begin{aligned} N^{4} \delta ^\mathrm{H} v_{x}^{4}-\lambda r N^{4} + \mu ^{43}\delta ^\mathrm{H}v_{x}^{4}&= 0 \end{aligned}$$
(A.26)
$$\begin{aligned} N^{4}\psi ^{'4}-\lambda N^{4} n^\mathrm{H} + \mu ^{43}\psi ^{'43}&= 0 \end{aligned}$$
(A.27)

The distortions in this case are

$$\begin{aligned} \begin{aligned}&d^{1}=0 \\&d^{3}=\frac{\mu ^{43}}{N^{3}+\mu ^{31}-\mu ^{43}}\varDelta ^\mathrm{HL} \\&d^{4}=0. \end{aligned} \end{aligned}$$
(A.28)

Three types: high-productivity workers pooled

Using the information of the binding self-selection constraints provided by numerical solution, the Lagrange function can be written as

$$\begin{aligned} \mathcal {L}= & {} N^1 [u(c^{1})+\delta ^\mathrm{L}v(x^{1})+\psi (1-y^{1})]\nonumber \\&+N^2 [u(c^{2})+\delta ^\mathrm{H}v(x^{2})+\psi (1-y^{2})]\nonumber \\&+\lambda \left[ \sum _{i=1}^{n} N^{i}(n^{i}y^{i}-c^{i}-rx^{i})-R \right] \nonumber \\&+\mu ^{21}\left[ u(c^{2})+\delta ^\mathrm{H}v(x^{2})+\psi (1-y^{2})- {u}(c^{1})\right. \nonumber \\&\left. -\delta ^\mathrm{H} {v}(x^{1})- {\psi }(1-y^{1})\right] \nonumber \\&+\mu ^{41}\left[ u(c^{4})+\delta ^\mathrm{H}v(x^{4})+\psi (1-y^{4})- {u}(c^{1})\right. \nonumber \\&\left. -\delta ^\mathrm{H} {v}(x^{1})- {\psi }\left( 1-\frac{n^\mathrm{L}}{n^\mathrm{H}}y^{1}\right) \right] \nonumber \\&+\mu ^{42}\left[ u(c^{4})+\delta ^\mathrm{H}v(x^{4})+\psi (1-y^{4})- {u}(c^{2})\right. \nonumber \\&\left. -\delta ^\mathrm{H} {v}(x^{2})- {\psi }\left( 1-\frac{n^\mathrm{L}}{n^\mathrm{H}}y^{2}\right) \right] \end{aligned}$$
(A.29)

The first-order conditions with respect \(c^{i}, x^{i}\) and \(y^{i}, i=1,2,4\) are given by

$$\begin{aligned} N^{1}u^{1}_{c}-\lambda N^{1}- \mu ^{21} u^{1}_{c}-\mu ^{41}u^{1}_{c}&= 0 \end{aligned}$$
(A.30)
$$\begin{aligned} N^{1}\delta ^\mathrm{L} v^{1}_{x}-\lambda r N^{1}- \mu ^{21} \delta ^\mathrm{H} v^{1}_{x}-\mu ^{41} \delta ^\mathrm{H}v^{1}_{x}&= 0\end{aligned}$$
(A.31)
$$\begin{aligned} -N^{1}\psi ^{1}_{y}+\lambda N^{1} n^\mathrm{L} + \mu ^{21} \psi ^{1}_{y}+\mu ^{41}\frac{n^\mathrm{L}}{n^\mathrm{H}}\psi ^{1}_{y}&= 0\end{aligned}$$
(A.32)
$$\begin{aligned} N^{2}u^{2}_{c}-\lambda N^{2}+ \mu ^{21} u^{2}_{c}-\mu ^{42} u^{2}_{c}&= 0\end{aligned}$$
(A.33)
$$\begin{aligned} N^{2}\delta ^\mathrm{H}v^{2}_{x}-\lambda r N^{2}+ \mu ^{21} \delta ^\mathrm{H}v^{2}_{x}-\mu ^{42} \delta ^\mathrm{H} v^{2}_{x}&= 0\end{aligned}$$
(A.34)
$$\begin{aligned} -N^{2}\psi ^{2}_{y}+\lambda N^{2} n^\mathrm{L}- \mu ^{21} \psi ^{2}_{y}+\mu ^{42}\frac{n^\mathrm{L}}{n^\mathrm{H}}\psi ^{2}_{y}&= 0\end{aligned}$$
(A.35)
$$\begin{aligned} -\lambda N^{4} + \mu ^{42} u^{4}_{c} + \mu ^{41} u^{4}_{c}&=0\end{aligned}$$
(A.36)
$$\begin{aligned} -\lambda r N^{4} + \mu ^{42} \delta ^\mathrm{H}v^{4}_{x} + \mu ^{41} \delta ^\mathrm{H}v^{4}_{x}&=0\end{aligned}$$
(A.37)
$$\begin{aligned} \lambda N^{4}n^\mathrm{H} - \mu ^{42} \psi ^{4}_{y} - \mu ^{41} \psi ^{4}_{y}&=0 \end{aligned}$$
(A.38)

Four types

Using the information of the binding self-selection constraints provided by numerical solution, the optimization problem can be written as

$$\begin{aligned} \mathcal {L}= & {} N^1 [u(c^{1})+\delta ^\mathrm{L}v(x^{1})+\psi (1-y^{1})\nonumber \\&+N^2 [u(c^{2})+\delta ^\mathrm{H}v(x^{2})+\psi (1-y^{2})\nonumber \\&+\lambda \left[ \sum _{i=1}^{n} N^{i}(n^{i}y^{i}-c^{i}-rx^{i})-R \right] \nonumber \\&+ \mu ^{31}\left[ u(c^{3})+\delta ^\mathrm{L}v(x^{3})+\psi (1-y^{3})- {u}(c^{1})-\delta ^\mathrm{L} {v}(x^{1})- {\psi }\left( 1-\frac{n^\mathrm{L}}{n^\mathrm{H}}y^{1}\right) \right] \nonumber \\&+\mu ^{32}\left[ u(c^{3})+\delta ^\mathrm{L}v(x^{3})+\psi (1-y^{3})- {u}(c^{2})-\delta ^\mathrm{L} {v}(x^{2})- {\psi }\left( 1-\frac{n^\mathrm{L}}{n^\mathrm{H}}y^{2}\right) \right] \nonumber \\&+\mu ^{42}\left[ u(c^{4})+\delta ^\mathrm{H}v(x^{4})+\psi (1-y^{4})- {u}(c^{2})-\delta ^\mathrm{H} {v}(x^{2})- {\psi }\left( 1-\frac{n^\mathrm{L}}{n^\mathrm{H}}y^{2}\right) \right] \nonumber \\&+\mu ^{43}\left[ u(c^{4})+\delta ^\mathrm{H}v(x^{4})+\psi (1-y^{4})- {u}(c^{3})-\delta ^\mathrm{L} {v}(x^{3})- {\psi }(1-y^{3})\right] \nonumber \\ \end{aligned}$$
(A.39)

The first-order conditions with respect to \(c^{i}, x^{i}\), and \(y^{i}\) for \(i=1,2,3,4\) are

$$\begin{aligned} N^{1}u^{1}_{c}-\lambda N^{1}- \mu ^{31} u^{1}_{c}&= 0\end{aligned}$$
(A.40)
$$\begin{aligned} N^{1}\delta ^\mathrm{L} v^{1}_{x}-\lambda r N^{1}- \mu ^{31} \delta ^\mathrm{L} v^{1}_{x}&= 0\end{aligned}$$
(A.41)
$$\begin{aligned} -N^{1}\psi ^{1}_{y}+\lambda N^{1} n^\mathrm{L}+\mu ^{31}\frac{n^\mathrm{L}}{n^\mathrm{H}}\psi ^{1}_{y}&= 0\end{aligned}$$
(A.42)
$$\begin{aligned} N^{2}u^{2}_{c}-\lambda N^{2}-\mu ^{32} u^{2}_{c}-\mu ^{42} u^{2}_{c}&= 0\end{aligned}$$
(A.43)
$$\begin{aligned} N^{2}\delta ^\mathrm{H}v^{2}_{x}-\lambda r N^{2}- \mu ^{32} \delta ^\mathrm{L}v^{2}_{x}-\mu ^{42} \delta ^\mathrm{H} v^{2}_{x}&= 0\end{aligned}$$
(A.44)
$$\begin{aligned} -N^{2}\psi ^{2}_{y}+\lambda N^{2} n^\mathrm{L}+\mu ^{32}\frac{n^\mathrm{L}}{n^\mathrm{H}} \psi ^{2}_{y}+\mu ^{42}\frac{n^\mathrm{L}}{n^\mathrm{H}}\psi ^{'2}_{y}&= 0\end{aligned}$$
(A.45)
$$\begin{aligned} -\lambda N^{3} + \mu ^{31} u^{3}_{c} + \mu ^{32} u^{3}_{c}-\mu ^{43} u^{3}_{c}&=0\end{aligned}$$
(A.46)
$$\begin{aligned} -\lambda r N^{3} + \mu ^{31} \delta ^\mathrm{L}v^{3}_{x} + \mu ^{32} \delta ^\mathrm{L}v^{3}_{x}-\mu ^{43} \delta ^\mathrm{H}v^{3}_{x}&=0\end{aligned}$$
(A.47)
$$\begin{aligned} \lambda N^{3}n^\mathrm {H} - \mu ^{31}\psi ^{3}_{y} - \mu ^{32}\psi ^{3}_{y}+\mu ^{43}\psi ^{3}_{y}&=0\end{aligned}$$
(A.48)
$$\begin{aligned} -\lambda N^{4} + \mu ^{42} u^{4}_{c} + \mu ^{43} u^{4}_{c}&=0\end{aligned}$$
(A.49)
$$\begin{aligned} -\lambda r N^{4} + \mu ^{42} \delta ^\mathrm{H}v^{4}_{x} + \mu ^{43} \delta ^\mathrm{H}v^{4}_{x}&=0\end{aligned}$$
(A.50)
$$\begin{aligned} \lambda N^{4}n^\mathrm{H} - \mu ^{42} \psi ^{4}_{y} - \mu ^{41} \psi ^{4}_{y}&=0 \end{aligned}$$
(A.51)

Appendix B: Additional results from the numerical simulations

See Tables 9, 10, 11, 12, 13 and 14.

Table 9 Lagrange multipliers and average tax rates for two-type model, maximin case
Table 10 Lagrange multipliers and average tax rates for two-type model in utilitarian case
Table 11 Lagrange multipliers and average tax rates for three-type model in maximin case
Table 12 Lagrange multipliers and average tax rates for three-type model in maximin case
Table 13 Lagrange multipliers and average tax rates for three-type model in utilitarian case
Table 14 Lagrange multipliers and average tax rates for three-type model in utilitarian case

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Ravaska, T., Tenhunen, S. & Tuomala, M. On the optimal lifetime redistribution and social objectives: a multidimensional approach. Int Tax Public Finance 25, 631–653 (2018). https://doi.org/10.1007/s10797-017-9473-0

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Keywords

  • Optimal taxation
  • Lifetime redistribution
  • Heterogeneous time preferences

JEL Classification

  • H21
  • H55
  • D71