On the design of an optimal non-linear tax/pension system with habit formation

Abstract

We study the optimal tax/pension design in a two-period model where individuals differ in both productivity and discount rates or projection bias and where their utility of the retirement period consumption is not independent of the earlier standard of living. We consider both welfarist and paternalistic social objectives. The paternalistic government attempts to correct the projection bias by using a higher discount factor. We derive general mathematical expressions that characterize optimal tax/pension design (marginal tax/subsidy rates). They suggest that the pattern of marginal labor income taxes depends on habit formation. Negative marginal labor income tax rates are possible. To gain a better understanding, we examine numerically the properties of an optimal lifetime redistribution policy with habit formation. We find support for non-linear tax/pension program in which some types of individuals are taxed while some are subsidized. The effect of changes in the degree of habit formation is explored in the numerical simulations as well as the implications of different degrees of correlation between skill and projection bias.

Appendices

Appendix A: Derivations of implicit taxes on savings

1.1 A.1 Derivation of (12a)–(12d)

Step 1: To rewrite the marginal rate of substitution $\frac {u_{c} + \delta^{i}\rho v_{c}}{\delta^{i}v_{x}}$ with help of the first order conditions, we first solve the numerator from (5). By adding and subtracting a term ∑ _j μ ^ji δ ⁱ ρv _c the first order condition with respect to c given in (5) can be written as

(21)

From this we can solve

$$ u_{c} + \delta^{i}\rho v_{c} = \frac{\lambda N^{i} + \sum_{j} \mu^{ji} ( \delta^{j} - \delta^{i} )\rho v_{c}}{N^{i} + \sum_{j} \mu^{ij} - \sum_{j} \mu^{ji}}. $$

(22)

Similarly we solve for the denominator of the marginal rate of substation. By adding and subtracting the term ∑ _j μ ^ji δ ⁱ v _x to the first order condition with respect to x given in (6), it can be written as

$$ N^{i}\delta^{i}v_{x} - \lambda N^{i}r + \sum_{j} \mu^{ij} \delta^{i}v_{x} - \sum_{j} \mu^{ji}\delta^{i}v_{x} - \sum _{j} \mu^{ji} \bigl( \delta^{j} - \delta^{i} \bigr)v_{x} = 0. $$

(23)

From this we can solve

$$ \delta^{i}v_{x} = \frac{\lambda N^{i}r + \sum_{j} \mu^{ji} ( \delta^{j} - \delta^{i} )v_{x}}{N^{i} + \sum_{j} \mu^{ij} - \sum_{j} \mu^{ji}}. $$

(24)

By combining (21) and (22) and making use of definition of the proportional difference in discount rates, $\Delta^{ji} = \frac {\delta ^{j} - \delta^{i}}{\delta^{i}}$, we can write

$$ \frac{u_{c} + \delta^{i}\rho v_{c}}{\delta^{i}v_{x}} = \frac{\lambda N^{i} + \delta^{i}\sum_{j} \mu^{ji}\Delta^{ji}\rho v_{c}}{\lambda N^{i}r + \delta^{i}\sum_{j} \mu^{ji}\Delta^{ji}v_{x}}. $$

(25)

Step 2: Manipulating equation to a useful form

The next trick is to multiply the both sides of (25) by term $\frac{\lambda N^{i}r + \delta^{i}\sum_{j} \mu^{ji}\Delta ^{ji}v_{x}}{ \delta^{i}\sum_{j} \mu^{ji}\Delta^{ji}v_{x}}$ . This gives us

$$ \frac{\lambda N^{i}r}{\delta^{i}\sum_{j} \mu ^{ji}\Delta ^{ji}v_{x}} \frac{u_{c} + \delta^{i}\rho v_{c}}{\delta^{i}v_{x}} + \frac {u_{c} + \delta^{i}\rho v_{c}}{\delta^{i}v_{x}} = \frac{\lambda N^{i}}{\delta^{i}\sum_{j} \mu^{ji}\Delta^{ji}v_{x}} + \frac{\delta^{i}\rho v_{c}}{\delta^{i}v_{x}}. $$

(26)

Rearranging terms gives us

$$ \frac{\lambda N^{i}r}{\delta^{i}\sum_{j} \mu ^{ji}\Delta ^{ji}v_{x}} \biggl[ \frac{u_{c} + \delta^{i}\rho v_{c}}{\delta ^{i}v_{x}} - \frac{1}{r} \biggr] - = - \frac{u_{c}}{\delta^{i}v_{x}}. $$

(27)

With this form we can solve for the marginal rate of substitution, $\frac{u_{c} + \delta^{i}\rho v_{c}}{\delta^{i}v_{x}}$, in a form given in (12a)–(12d).

1.2 A.2 Derivation of (20)

Paternalistic case follows a similar procedure as in a welfarist case, so details of the derivations are omitted. The terms to be added and subtracted to first order conditions are now ∑ _j μ ^ij δ ^g ρv _c −∑ _j μ ^ji δ ^g ρv _c and ∑ _j μ ^ij δ ^g v _x −∑ _j μ ^ji δ ^g v _x . The term that is used to multiply equation in step 2 is

$$\frac{\lambda N^{i}r - \delta^{g}\sum_{j} \mu^{ij}\Delta ^{ig}v_{x} + \delta^{g}\sum_{j} \mu^{ji}\Delta^{jg}v_{x}}{ - \delta ^{g}\sum_{j} \mu^{ij}\Delta^{ig}v_{x} + \delta^{g}\sum_{j} \mu ^{ji}\Delta^{jg}v_{x}}. $$

1.3 A.3 Derivation of (∗) in Footnote 15

Derivation proceeds similarly to the earlier cases. The terms to be added and subtracted to the first order conditions are N ⁱ δ ⁱ ρv _c −∑ _j μ ^ji δ ⁱ ρv _c and N ⁱ δ ⁱ v _x −∑ _j μ ^ji δ ⁱ v _x . In step 2, the multiplier to be used is

$$\frac{\lambda N^{i}r - \delta^{i}( N^{i}\Delta^{gi} - \sum_{j} \mu^{ji}\Delta^{ji} )v_{x}}{ - \delta^{i}( N^{i}\Delta^{gi} - \sum_{j} \mu^{ji}\Delta^{ji} )v_{x}}. $$

Appendix B: Numerical simulations

2.1 B.1 Procedure

Numerical simulation was carried out with the Matlab program. The function used (fmincon) solves the optimum of a multivariable function with equality and inequality constraints that may be linear or non-linear. It also determines which of the constraints are binding. The same procedure is applied to all cases considered in this paper.

Note that as the optimization function also allowed slack constraints, we were not restricted to a priori assumptions on the binding self-selection constraints. Thus, we included all possible constraints in the optimization procedure and simply determined the binding constraints with the help of numerical solutions.

Multidimensional screening problems are numerically challenging to solve, as discussed in Judd and Su (2006). We also encountered some difficulties with the solvability of the problem, as the matrix including the constraints with opposite self-selection constraints is very close to being singular. These difficulties were met during the sensitivity analysis; with some combinations of parameters the problem was not solvable or gave irrational values.

2.2 B.2 Parameterization

The distribution of the economy was chosen to be uniform merely for the simplicity and comparability for the following cases. The discount factors, δ ^L and δ ^H, were set as 0.6 and 0.8, while δ ^g was set equal to 1. These values are in line with e.g. Cremer et al. (2009) who performed similar numerical calculations. The wage rates reflecting the productivities were chosen to be 2 and 3, respectively. The magnitude of the wage rates was determined by the solvability of the problem; with these wage rates, the labor supply decisions, restricted to lie between 0 and 1, were at reasonable levels.

2.3 B.3 Sensitivity analysis

We have considered the sensitivity of the analysis with the help of varying (i) the degree of habit formation, which allows us to compare the cases with and without habit effect, and (ii) the correlation between the characteristics of the individuals, skill level and discount factor. With the some combinations of the parameters there is a failure to find the optimum, but it is due to technical problems in the optimization algorithm.

The first part, sensitivity with respect to degree of habit formation, is presented in the main text. The results considering sensitivity with respect to correlation between the two unobservable features follow here.