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Published in:

30-08-2017

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

Authors: Terhi Ravaska, Sanna Tenhunen, Matti Tuomala

Published in: International Tax and Public Finance | Issue 3/2018

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## 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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Appendix
Available only for authorised users
Footnotes
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.

Literature
Atkinson, A., & Stiglitz, J. (1976). The design of tax structure: Direct versus indirect taxation. Journal of Public Economics, 6, 55–75. CrossRef
Bossert, W. (1995). Redistribution mechanisms based on individual characteristics. Mathemathical Social Sciences, 29(1), 1–17. CrossRef
Choné, P., & Laroque, G. (2010). Negative marginal tax rates and heterogeneity. American Economic Review, 100(5), 2532–47. CrossRef
Cremer, H., De Donder, P., Maldonado, D., & Pestieau, P. (2009). Forced saving, redistribution and non-linear social security scheme. Southern Economic Journal, 76(1), 86–98. CrossRef
Diamond, P. A. (2003). Taxation incomplete markets and social security. The 2000 Munich lectures. Cambridge, MA: MIT Press.
Diamond, P. A., & Spinnewijn, J. (2011). Capital income taxes with heterogeneous discount rates. American Economic Journal: Economic Policy, 3(4), 52–76.
Fleurbaey, M. (1994). On fair compensation. Theory and Decision, 36, 277–307. CrossRef
Fleurbaey, M., Leroux, M.-A., & Ponthiere, G. (2014). Compensating the dead. Journal of Mathematical Economics, 51(March), 28–41. CrossRef
Fleurbaey, M., & Maniquet, F. (2006). Fair income taxation. The Review of Economic Studies, 73(1), 55–83. CrossRef
Golosov, M., Troshkin, M., Tsyvinski, A., & Weinzierl, M. (2013). Preference heterogeneity and optimal capital income taxation. Journal of Public Economics, 97, 160–175. CrossRef
Lockwood, B. B., & Weinzierl, M. (2015). De Gustibus non est taxandum: Heterogeneity in preferences and optimal redistribution. Journal of Public Economics, 124, 74–80. CrossRef
Mirrlees, J. (1971). An exploration in the theory of optimum income taxation. The Review of Economic Studies, 38(2), 175–208.
Roemer, J. E. (1998). Equality of opportunity. Harvard: Harvard University Press.
Saez, E. (2002). The desirability of commodity taxation under non-linear income taxation and heterogeneous tastes. Journal of Public Economics, 83, 217–230. CrossRef
Sandmo, A. (1993). Optimal redistribution when tastes differ. Finanz Archiv, 50(2), 149–163.
Stern, N. (1982). Optimum taxation with errors in administration. Journal of Public Economics, 17, 181–211. CrossRef
Stiglitz, J. (1982). Self-selection and pareto efficient taxation. Journal of Public Economics, 17, 213–240. CrossRef
Tarkiainen, R., & Tuomala, M. (1999). Optimal nonlinear income taxation with a two-dimensional population: A computational approach. Computational Economics, 13, 1–16. CrossRef
Tarkiainen, R., & Tuomala, M. (2007). On optimal income taxation with heterogeneous work preferences. International Journal of Economic Theory, 3, 35–46. CrossRef
Tenhunen, S., & Tuomala, M. (2010). On optimal lifetime redistribution policy. Journal of Public Economic Theory, 12(1), 171–198. CrossRef
Tenhunen, S., & Tuomala, M. (2013). On the design of an optimal non-linear tax/pension system with habit formation. International Tax and Public Finance, 20(3), 485–512. CrossRef
Van de Gaer, D. (1993). Equality of opportunity and investments in human capital. Ph.D. Thesis, Katholieke Universiteit Leuven.
Metadata
Title
On the optimal lifetime redistribution and social objectives: a multidimensional approach
Authors
Terhi Ravaska
Sanna Tenhunen
Matti Tuomala
Publication date
30-08-2017
Publisher
Springer US
Published in
International Tax and Public Finance / Issue 3/2018
Print ISSN: 0927-5940
Electronic ISSN: 1573-6970
DOI
https://doi.org/10.1007/s10797-017-9473-0

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