2.1 Economic and demographic assumptions

The individual is age t ₀ when the problem starts and he can live up to age ω. The probability that he survives from age t ₀ to age t is denoted by $p_{t_{\setnum{0}} \comma t} $. It is assumed that $p_{t_{\setnum{0}} \comma t} $ is continuous, decreases with time, and eventually converges to zero as t → ω. The utility of consumption is represented by an increasing and concave function u(c) with u′(c) > 0, u″(c) < 0, $\mathop {{\rm lim}}\limits_{c \to \setnum{0}} u\prime \lpar c\rpar \equals \infty $, and $\mathop {{\rm lim}}\limits_{c \to \infty } u\prime\lpar c\rpar \equals 0$. Time preferences are taken into account by discounting utility at a continuous rate β. The combined discount from time and mortality is denoted by

(2)$$f\hskip 1\lpar t\rpar \equals {\rm e}^{ \minus \beta \lpar t \minus t_{\setnum{0}} \rpar } p_{t_{\setnum{0}} \comma t}.$$

All economic assumptions are expressed in real terms. Savings grow at a real risk-free rate r > 0. The wealth process is denoted by W_t and it is assumed that the individual starts the problem with no initial savings, i.e. $W_{t_{\setnum{0}} } \equals 0$. Borrowing is not allowed. Before retirement, pre-tax income is a continuous function denoted by y_t. The individual retires at age R, which is assumed to be 65 years old. After retirement, the individual receives a Social Security pension with annual payments of SS > 0. He may also receive annual income y_R ⩾ 0 from an employer pension or another source of annuity payments, for a total of y_t=y_R + SS per year. The solution considers only the case where expected after-tax income declines at the time of retirement.

2.2 Taxation

Before retirement, income is subject to a payroll tax rate π. At all times, federal income taxes based on the United States tax bracket system apply.Footnote ⁸ For k = 0, …, K, the brackets are denoted by [B_k,B_{k +1}) and the marginal tax rate within bracket k is τ_k. Tax rates after retirement are subject to a one-time multiplicative shock θ that applies to all marginal tax rates. To represent this risk, the model allows choosing across the entire family of discrete probability distribution. Let N be the number of possible states, there is a probability p_i with i = 1, …, N that θ = θ_i and $\tau _{k}^{i} \equals \theta _{i} \tau _{k} $. The case without tax risk can be viewed as a special case of this model with N = 1 and θ₁ = 1. For the remainder of this paper, all functions of τ_k should be interpreted as random variables after retirement although the notation is not differentiated.

To compute taxable income, the standard deduction and personal exemption must be taken into account. The sum of these two components is denoted by E before age 65 and by E_R after age 65 (there is an increase in the standard deduction at age 65). In the traditional case, it must also be recognized that taxable income is reduced by contributions before retirement and increased by withdrawals after retirement. The contributions or savings before retirement are denoted by s_t where s_t=y_t−c_t − tax_t(s_t) and c_t is consumption. A negative s_t represents a withdrawal. Let SS_t^tx denote the taxable portion of Social Security benefits, the taxable income function is given by

(3)$$y_{t}^{{\rm tx}} \lpar s_{t} \rpar \equals \left\{ {\matrix{ {y_{t} \minus \alpha s_{t} \minus E\comma } \hfill \tab {t \lt R\comma } \hfill \cr {y_{R} \plus SS_{t}^{{\rm tx}} \lpar s_{t} \rpar \minus \alpha s_{t} \minus E_{R} \comma } \hfill \tab {t\geqslant R.} \hfill \cr} } \right.$$

Accordingly, for k = 0, …, K, the tax function is

(4)$${\rm tax}_{t} \lpar s_{t} \rpar \equals \pi y_{t} \cdot 1\lpar t \lt R\rpar \plus \tau _{k} y_{t}^{{\rm tx}} \lpar s_{t} \rpar \plus G_{k} \comma \quad \quad B_{k} \leqslant y_{t}^{{\rm tx}} \lpar s_{t} \rpar \lt B_{k \plus \setnum{1}} \comma $$

where

(5)$$G_{k} \equals \mathop{\sum}\limits_{j \equals \setnum{1}}^{k \minus \setnum{1}} {\tau _{j} } \lpar B_{j \plus \setnum{1}} \minus B_{j} \rpar \minus \tau _{k} B_{k}.$$

It should be noted that some of the features of the tax system are not included in the model because they do not affect the solution in most cases illustrated here. For instance, the model does not limit the amount of tax deductible contributions because the optimal savings in all our illustrations are below the 401(k)’s $16,500 limit. In addition, the 10% penalty tax for early withdrawals before age 59½ is not incorporated because the model does not feature shocks that would trigger them. This eventuality is discussed later in Section 6.3.

The model is completed by specifying SS_t^tx(s_t), the taxable portion of Social Security benefits. This is a fairly complex formula. First, a provisional income measure PI_t ≡ PI(s_t) is defined by adding half of Social Security benefits to other sources of income (including withdrawals from traditional accounts, but not Roth accounts), yielding:

(6)$${\rm Provisional}\;{\rm Income}_{t} \equals {\rm PI}\lpar s_{t} \rpar \equals y_{R} \plus {{{\rm SS}} \over 2} \minus \alpha s_{t} \comma \quad \quad t\geqslant R.$$

Once PI_t is computed, it has to be compared to two ‘base amounts’ X ¹ and X ². Currently, these base amounts are X ¹ = $25,000 and X ² = $34,000 for singles; the corresponding measures are $32,000 and $44,000 for those married filing jointly. These amounts are not indexed for inflation. Given that the rest of the model is expressed in real terms, these thresholds have to be adjusted for inflation (denoted by i) and this creates cohort effects in the model. For example, for somebody retiring at age 65 in 2010, the first base amount in real terms is $25,000. For somebody currently age 45 in 2010, this same base amount will be $13,720 when they retire in 20 years (assuming i = 3%), making it more likely that they will be taxed. For a given individual, the base amounts will also decrease by e^−it in real terms after t years of retirement. Thus, let a be the current age in 2010, the base amounts in real terms can be expressed by

(7)$$X_{t}^{\setnum{1}} \equals X^{\setnum{1}} {\rm e}^{ \minus {\rm i}\lpar t \minus a\rpar } \quad \quad {\rm and}{\kern 1pt} \quad \quad X_{t}^{\setnum{2}} \equals X^{\setnum{2}} {\rm e}^{ \minus {\rm i}\lpar t \minus a\rpar }.$$

If the provisional income PI_t is less than the first base amount $X_{t}^{\setnum{1}} $, none of the Social Security benefits are taxable. If PI_t is greater than the first base amount, but lower than the second, 50% of the excess PI_t−X _t¹ is taxable income.Footnote ⁹ If PI_t is greater than the second base amount, then the taxable portion is 50%(X _t²−X _t¹) + 85%(PI_t−X _t²), subject to a maximum of 85% of Social Security benefits. Thus, depending on the level of PI_t, there are four different formulas that can apply: 0, 50%(PI_t−X_t ¹), 50%(X _t²−X _t¹) + 85%(PI_t−X _t²), and 85% SS. To summarize, these four cases can be embedded into one linear structure:

(8)$${\rm SS}_{t}^{{\rm tx}} \lpar s_{t} \rpar \equals M_{h} {\rm PI}\lpar s_{t} \rpar \plus H_{t\comma h} \comma \quad \quad B_{t\comma h}^{S} \leqslant {\rm PI}\lpar s_{t} \rpar \lt B_{t\comma h \plus \setnum{1}}^{S} \comma $$

where

(9)

The key notation to remember for the rest of the analysis is M_h, which represents the marginal rate of inclusion of Social Security benefits in taxable income. From equations (6) and (8), a dollar withdrawn from a traditional account increases the taxable portion of Social Security benefits by M_h and triggers an additional tax of τ_kM_h. Thus, M_h can be viewed as a factor that magnifies the marginal tax rate from τ_k to τ_k(1+M_h). For the remainder of this paper, τ_k(1+M_h) will be referred to as the effective marginal tax rate. For example, if τ_k = 15% and M_h = 85, withdrawals from traditional accounts are effectively taxed at a rate of 15% (1 + 85%) = 27.75%.

To put this in perspective, recall that the basis for favoring traditional accounts is that marginal tax rates after retirement should be lower than before retirement. Once the taxation of Social Security benefits is taken into account, this logic may no longer hold as it is possible to have lower income after retirement, but higher tax rates. This issue does not necessarily affect everyone, those with either very low or very high incomes are not impacted because their inclusion rate M_h is zero. It should be kept in mind that the composition of this group varies across cohorts because the base amounts change in real terms. Initially, those with higher incomes are subject to the 50% or 85% inclusion rates. Eventually, they will see their M_h decline to zero when their taxable benefits reach 85% SS. On the other hand, those with lower incomes start with M_h = 0 but ultimately see their inclusion rates increase to 50% and 85% as the base amounts decrease in real terms.

2.3 Optimization problem

Combining these assumptions, the individual's problem is to choose consumption to maximize his expected utility, subject to a budget constraint and borrowing constraints. Thus, the optimization problem is

(10)$$\mathop {{\rm max}}\limits_{c \gt \setnum{0}} E\left[ {\int_{t_{\setnum{0}} }^{\omega } {f\hskip 1\lpar t\rpar u\lpar c_{t} \rpar {\rm d}t} } \right]$$

such that

(11)$${\rm d}W_{t} \equals \left[ {W_{t} r \plus y_{t} \minus c_{t} \minus {\rm tax}_{t} \lpar s_{t} \lpar c_{t} \rpar \rpar } \right]{\rm d}t$$

and

(12)$$W_{t} \geqslant 0\quad \forall t\comma \quad \quad W_{t_{\setnum{0}} } \equals 0\comma $$

where tax_t(s_t) is given in (4) and s_t(c_t) by

(13)$$s_{t} \lpar c_{t} \rpar \equals {{y_{t} \minus {\rm tax}_{t} \lpar 0\rpar \minus c_{t} } \over {1 \minus \alpha \tau _{k} \lpar 1 \plus M_{h} \rpar }}$$

for $B_{k} \leqslant y_{t}^{{\rm tx}} \lpar s_{t} \rpar \lt B_{k \plus \setnum{1}} $ and $B_{t\comma h}^{S} \leqslant {\rm PI}\lpar s_{t} \rpar \lt B_{t\comma h \plus \setnum{1}}^{S} $ with $y_{t}^{{\rm tx}} \lpar s_{t} \rpar $ given in (3) and PI(s_t) given in (6).

2.4 Discontinuity points for τ_k and M_h

The rates τ_k and M_h in this model are step functions of consumption and these discontinuities affect the structure of the solution in the next section. To facilitate the exposition of the solution, the notations C_k and C_h are introduced to identify the levels of consumption where respectively the marginal tax rate jumps from τ_{k− 1} to τ_k and the rate of inclusion of Social Security benefits jumps from M_{h− 1} to M_h. We can solve for C_k by setting taxable income equal to the next tax bracket (i.e. $y_{t}^{{\rm tx}} \lpar s\lpar C_{k} \rpar \rpar \equals B_{k} $) and inverting C_k from equation (13) as follows:Footnote ¹⁰

(14)$$C_{k} \equals \left\{ {\matrix{ {y_{t} \lpar 1 \minus \pi \minus 1\sol \alpha \rpar \plus B_{k} \lpar 1\sol \alpha \minus \tau _{k} \rpar \minus G_{k} \plus E\sol \alpha \comma } \hfill \tab {t \lt R\comma } \hfill \cr {y_{R} \left( {1 \minus {1 \over \alpha }} \right) \plus {\rm SS}\left( {1 \minus {{M_{h} \sol 2} \over {\alpha \lpar 1 \plus M_{h} \rpar }}} \right) \plus B_{k} \left( {{1 \over {\alpha \lpar 1 \plus M_{h} \rpar }} \minus \tau _{k} } \right) \minus G_{k} \plus {{ \minus H_{t\comma h} \plus E_{R} } \over {\alpha \lpar 1 \plus M_{h} \rpar }}\comma } \hfill \tab {t\geqslant R.} \hfill \cr} } \right.$$

Similarly, C_h is derived by setting the provisional income in equation (6) equal to the next threshold $B_{t\comma h}^{S} $ to obtain:

(15)$$\eqalign{ C_{h} \equals \tab y_{R} \left( {1 \minus {1 \over \alpha }} \right) \plus {\rm SS}\left( {1 \plus {{\tau _{k} \alpha \minus 1} \over {2\alpha }}} \right) \plus\cr \tab B_{t\comma h}^{S} \times \left( {{1 \over \alpha } \minus \tau _{k} \lpar 1 \plus M_{h} \rpar } \right) \minus G_{k} \minus \tau _{k} \lpar H_{t\comma h} \minus E^{R} \rpar. \cr} $$

3.1 Lagrangian and dual approach

Solving the problem with conventional methods can be challenging because we need to prove that the borrowing constraint is satisfied everywhere. Without restricting the model's assumptions, this is a difficult task because the optimal consumption function provides little insight into the sign of the wealth process. In this context, it is useful to turn to the dual approach suggested in Lachance (Reference Lachance2012) and extend it to include discontinuities and risks. Dual approaches are used with problems that are difficult to solve in their primal form, but are easier to handle in their equivalent dual form. In this case, the advantage is that we do not need to prove directly that $W_{t}\ast \geqslant 0$ everywhere and this allows us to get a simpler condition.

As with conventional optimization methods, to prove that a solution is optimal the problem's Lagrangian must be derived first. Appendix A.1 describes how the standard Lagrangian is constructed and rewrites it in a form suitable for the dual approach as follows:

(16)$$L \equals E\left[ {\int_{t_{\setnum{0}} }^{\omega } {\lpar f\hskip 1\lpar t\rpar u\lpar c_{t} \rpar \plus X\lpar t\rpar {\rm e}^{ \minus r\lpar t \minus t_{\setnum{0}} \rpar } s_{t} \lpar c_{t} \rpar \rpar {\rm d}t} } \right].$$

The process X(t) is just a transformation of the original Lagrange multipliers; it must be decreasing during the binding periods and equal to a constant λ during the non-binding period [T ₁, T ₂]. With tax risk after retirement, a different constant λ_i applies in each state i. By construction, the constants are related by $\lambda \equals \sum _{i \equals \setnum{1}}^{N} p_{i} \lambda _{i} $ and can be interpreted as the marginal utility of wealth. Less technically, these conditions on X(t) mean that expected utility cannot be improved by saving an additional dollar at time t and spending it later (and vice-versa).

The first step of the dual approach is to find $c_{t}\ast $ that maximizes L, which is a simple unconstrained maximization problem detailed in Section 3.2. The second step of the dual approach is to find X(t) that minimizes L, which in our setup is equivalent to:

• • Solving for the constants λ and λ_i that satisfy the budget constraint equations W(λ, λ_i) = 0, i = 1, …, N and the condition $\lambda \equals \sum _{i \equals \setnum{1}}^{N} p_{i} \lambda _{i} $.

• • If there is more than one constant λ or λ_i that satisfies the budget constraint, the highest one is optimal.

A solution that satisfies these criteria maximizes utility and satisfies all the problem's constraints. Appendix A.2 details the equation for the budget constraint W(λ, λ_i) = 0 and Appendix A.3 explains how the bisection method can be employed to solve for the constants λ and λ_i.

3.2 Optimal consumption and issues with discontinuities

To find $c_{t}\ast $ that maximizes L unconditionally when X(t) = λ, the standard procedure is to derive the first-order condition (F.O.C.) as follows:

(17)$$\eqalign{ \tab L\prime\lpar c_{t} \rpar \equals f\hskip 1\lpar t\rpar u\prime\lpar c_{t} \rpar \minus {{\lambda {\rm e}^{ \minus r\lpar t \minus t_{\setnum{0}} \rpar } } \over {1 \minus \alpha \tau _{k} \lpar 1 \plus M_{h} \rpar }}\comma \cr \tab {\rm F}{\rm.O}{\rm.C}{\rm.}\colon \quad L\prime\lpar c_{t}\ast \rpar \equals 0\quad \Rightarrow \quad c_{t}\ast \lpar \lambda \rpar \equals u\prime^{ \minus \setnum{1}} \left( {{{\lambda {\rm e}^{ \minus r\lpar t \minus t_{\setnum{0}} \rpar } } \over {f\hskip 1\lpar t\rpar \lpar 1 \minus \alpha \tau _{k} \lpar 1 \plus M_{h} \rpar \rpar }}} \right). \cr} $$

The first component in L′(c_t) measures the marginal benefit of consumption and the second one the opportunity cost of saving where s _t′(c_t) = −1/(1 − ατ_k(1+M_h)). With traditional accounts, savings do not decrease one-for-one with consumption because taxes create an additional reward (or penalty for withdrawals). In that case, higher taxes translate into higher savings before retirement and lower withdrawals after retirement.

When L′(c_t) is continuous, L″(c_t) < 0 and there is a unique solution to the F.O.C. $L\prime\lpar c_{t}\ast \rpar \equals 0$. When τ_k and M_h enter L′(c_t), this cannot be taken as granted because the discontinuities raise new technical issues: (1) the F.O.C. may not have a solution $c_{t}\ast $ for every λ, (2) the F.O.C. can have more than one solution $c_{t}\ast $ for a given λ, and (3) circularity, i.e. τ_k is a function of $c_{t}\ast $ and $c_{t}\ast $ is a function of τ_k. Note that these discontinuity issues can also arise in retirement problems where retirement savings are rewarded (or withdrawals penalized) and the reward/penalty can suddenly change. For example, this would be the case if an employer offers a 50% match on employee contributions, but stops the match for contributions above 6%.

For the first issue, recall that Section 2.4 defined the discontinuity points C_k and C_h where τ_k and M_h jump. Here, the notation C will stand for any of these points. The problem is that it is not possible to solve for c in L′(c) = 0 when at a discontinuity point C we have L′(C⁻) > 0 and L′(C) < 0. Actually, with L″(c) < 0, C is locally optimal because L(c) increases up to C and decreases thereafter. Since the equation for L′(C) in (17) is a function of λ, it will be convenient to rewrite the condition in terms of a range of values for λ as follows:

(18)$$L\prime\lpar C \minus \rpar \gt 0\quad {\rm and}\quad L\prime\lpar C\rpar \lt 0\;{\rm is}\;{\rm equivalent}\;{\rm to}\;\rmLambda _{t} \lpar C\rpar \lt \lambda \lt \rmLambda _{t} \lpar C^{ \minus } \rpar \comma $$

where the function

(19)$$\rmLambda _{t} \lpar c\rpar \equals u&#x0027;\lpar c\rpar f\hskip 1\lpar t\rpar {\rm e}^{r\lpar t \minus t_{\setnum{0}} \rpar } \lpar 1 \minus \alpha \tau _{k} \lpar 1 \plus M_{h} \rpar \rpar $$

is simply introduced to express the solution more compactly later on.Footnote ¹¹ Outside these values for λ, the standard F.O.C. solution in (17) applies. This approach with brackets for λ offers a simple test to determine when an algorithm should use the standard F.O.C. solution in (17) and when it should use the discontinuity points C. It has also the advantage of addressing the circularity issue: for a given λ, we know which τ_k or M_h to use in $c_{t}\ast \lpar \lambda \rpar $.

The second problem that we can encounter is having more than one locally optimal solution. This happens when L′(c_t) increases at a discontinuity, i.e. in the uncommon scenario where the effective marginal tax rate decreases with consumption. In this model, this occurs at the point C_{h =3} where taxable Social Security benefits reach the 85% maximum and M_h drops from 85% to 0%. Specifically, when $L&#x0027;\lpar C_{h \equals \setnum{3}}^{ \minus } \rpar \lt 0$ and L′(C_{h =3}) > 0, there are two locally optimal solutions c ₁<C_{h =3} and c ₂>C_{h =3} (with h<3 for c ₁ and h = 3 for $c_{\setnum{2}} $). The globally optimal one maximizes L(c_t) and to express this more systematically, we solve for a point $\lambda \equals \bar{\rmLambda }$ such that the individual is indifferent between c ₁ and c ₂. It is possible to show that c ₁ is optimal for all $\lambda \gt \bar{\rmLambda }$, c ₂ is optimal for all $\lambda \lt \bar{\rmLambda }$, and the solution jumps between c ₁ and c ₂ when $\lambda \equals \bar{\rmLambda }$.Footnote ¹²Figure 1 makes this result more intuitive by illustrating the solution graphically for the cases $\lambda \lt \bar{\rmLambda }$, $\lambda \equals \bar{\rmLambda }$, and $\lambda \gt \bar{\rmLambda }$.

Figure 1. Illustration of jump in optimal consumption (optimal c_t maximizes L(c_t)).

Combining these results, the optimal solution in the non-binding period after retirement t∊[R, T ₂) can be condensed with:

(20)$$c_{t}\ast \lpar \lambda \rpar \equals \left\{ {\matrix{ {u&#x0027;^{ \minus \setnum{1}} \left( {{{\lambda e^{ \minus r\lpar t \minus t_{\setnum{0}} \rpar } } \over {f\hskip 1\lpar t\rpar \lpar 1 \minus \alpha \tau _{k} \lpar 1 \plus M_{h} \rpar \rpar }}} \right)\comma } \hfill \tab {\rm max} \left[ {\rmLambda _{t} \lpar C_{k \plus \setnum{1}}^{ \minus } \rpar \comma \rmLambda _{t} \lpar C_{h \plus \setnum{1}}^{ \minus } \rpar } \right]\leqslant \hfill \cr\tab\lambda \leqslant {\rm min} \left[ {\rmLambda _{t} \lpar C_{k} \rpar \comma \rmLambda _{t} \lpar C_{h} \rpar } \right]\comma \hfill \cr {C_{h} \comma } \hfill \tab {\rmLambda _{t} \lpar C_{h} \rpar \lt \lambda \lt \rmLambda _{t} \lpar C_{h}^{ \minus } \rpar \comma \quad h \lt 3}\comma \hfill \cr {C_{k}\comma } \hfill \tab {\rmLambda _{t} \lpar C_{k} \rpar \lt \lambda \lt \rmLambda _{t} \lpar C_{k}^{ \minus } \rpar \comma } \hfill \cr} } \right. $$

for $\lambda \geqslant \bar{\rmLambda }$ and

(21)$$c_{t}\ast \lpar \lambda \rpar \equals \left\{ {\matrix{ {u\prime^{ \minus \setnum{1}} \left( {{{\lambda {\rm e}^{ \minus r\lpar t \minus t_{\setnum{0}} \rpar } } \over {f\hskip 1\lpar t\rpar \lpar 1 \minus \alpha \tau _{k} \rpar }}} \right)\comma } \hfill \tab {\rmLambda _{t} \lpar C_{k \plus \setnum{1}}^{ \minus } \rpar \leqslant \lambda \leqslant \rmLambda _{t} \lpar C_{k} \rpar \comma } \hfill \cr {C_{k\comma } } \hfill \tab {\rmLambda _{t} \lpar C_{k} \rpar \lt \lambda \lt \rmLambda _{t} \lpar C_{k}^{ \minus } \rpar \comma } \hfill \cr} } \right.$$

for $\lambda \lt \bar{\rmLambda }$. Before retirement, the points C_h related to Social Security do not apply and the solution in the non-binding period [T ₁,R) is given by (21) for all λ. In the pure Roth case (α = 0), only the top part of the solution applies. In other words, working with Roth accounts in optimization problems is much easier than working with traditional accounts.

If the utility function takes the standard power utility form $u\lpar c\rpar \equals c^{\setnum{1} \minus \gamma } \sol \lpar 1 \minus \gamma \rpar $ with γ≠1, the top portion of the solution in (20) and (21) becomes:

(22)$$c_{t}\ast \lpar \lambda \rpar \equals \left( {{{\lambda {\rm e}^{\lpar \beta \minus r\rpar \lpar t \minus t_{\setnum{0}} \rpar } } \over {p_{t_{\setnum{0}} \comma t} \lpar 1 \minus \alpha \tau _{k} \lpar 1 \plus M_{h} \rpar \rpar }}} \right)^{ \minus \setnum{1}\sol \gamma }.$$

Appendix B shows how this equation can be combined with a flexible set of assumption for mortality and income to express the budget constraint W(λ, λ_i) = 0 as a series of closed-form equations.

3.3 Comparison with dynamic programming

In the previous literature, dynamic programming techniques have been the tool of choice for solving life-cycle models with risks and constraints, and it can be useful to contrast them with the approach described in this section. With dynamic programming, the problem is to solve the Bellman equation $V\lpar W_{t} \rpar \equals \mathop {{\rm max}}\limits_{c_{t} } \lsqb u\lpar c_{t} \rpar \plus \beta p_{t} E\lsqb V_{t \plus \setnum{1}} \lpar W_{t \plus \setnum{1}} \rpar \rsqb \rsqb $ subject to the budget constraint in (11), which is essentially a discretized version of our earlier problem of maximizing L(c_t). Since the functional form for the value function V_{t +1}(W_{t +1}) is not known, it must be estimated numerically by backward induction for a set of values, and then interpolated between these points. With this interpolated function, the simplest approach to solve the Bellman equation is to test every possible value of c_t: this is a slow process and has limited precision unless a fine grid is used. The circularity issue mentioned previously also makes the process more time consuming.

Although several numerical techniques can be used to perform a more efficient search for optimal consumption, their application is not straightforward with discontinuities. For instance, most assume the existence of a single optimum or solution within a given interval, but Figure 1 illustrated that this premise can fail. Furthermore, techniques based on first-order-conditions or related Euler equations have to be modified because the solution in (20) and (21) showed that several cases have to be considered. In other words, applying these techniques to problems with discontinuities requires some adjustments and an analytical process such as the one in Section 3.2 can be followed to determine the nature of the changes.

The technique used in this paper is particularly interesting because the solution is based on a known budget constraint instead of an unknown value function, which eliminates the need for interpolation and leads to exact values. In addition, it is not necessary to numerically optimize $c_{t}\ast $ as the solution is given in (20) and (21). Arguably, the approach suggested here is limited in the sense that it is based on a one-time risk, but the same concepts could be extended to multiple risky periods. With M possible risky paths, the crux of the problem would be to solve a system of M equations (the budget constraints, possibly as a series of closed-form expressions) and M unknowns λ₁, …, λ_M.Footnote ¹³ If this system can be solved in a reasonable amount of time, then the approach can be an interesting alternative to dynamic programming. With a one-time risk, the problem proved to be particularly manageable as it can be reduced to a one-equation-in-one-unknown problem that can be solved with the bisection method.

6.1 Breakeven increase in future tax rates

Since an increase in future tax rates would favor Roth accounts, it is interesting to ask: What is the magnitude of the increase needed to change the traditional account recommendation? To answer that question, recall that the model in Section 2 allows for a multiplicative increase θ that applies to all marginal tax rates after retirement. Using this framework, we solve for the breakeven increase θ that would make the individual indifferent between traditional and Roth accounts and present the results in the second column of Table 2. Note that the breakeven rates are negative for those who initially prefer Roth accounts. The table indicates ‘N/A’ for those who pay no (or very little) taxes after retirement since changes in future marginal tax rates are not an issue for them, they will prefer traditional accounts no matter what. Future tax changes, however, are relevant for those with a college degree and in a few of the high-school cases. Table 2 shows that for them, breakeven rates range between 4% and 41%. Those with high pensions have low breakeven rates and can be affected by even small changes in tax rates. In contrast, those with a college degree and no pension income are less sensitive and would require a more substantial increase of 41% (age 45 in 2010) or 33% (age 25 in 2010) to justify the switch to Roth accounts. To put this in perspective, a 33% increase would change the marginal tax rates in Table 1 to 0%, 13%, 20%, 33%, 37%, 44%, and 47%.

Although Roth accounts are immune to increases in future marginal tax rates, it should be acknowledged that they are subject to different risks. For example, a change in the tax law could include withdrawals from Roth accounts in the taxation of Social Security benefits. To illustrate the magnitude of this issue, the third column of Table 2 gives the resulting increase in the present value of taxes for those who save with Roth accounts. This amount can be quite substantial, reaching up to $18,443. If this change was expected with certainty, Roth accounts would never be optimal in our illustrations. In addition, Roth accounts would lose some of their appeal for younger cohorts if Social Security's base amounts become indexed for inflation. Of course, substantial tax reforms would also affect the comparison, for example Kotlikoff et al. (Reference Kotlikoff, Marx and Rapson2008) evaluate the impact of a change to a consumption tax. Unfortunately, most strategies for retirement savings are not truly risk-free as long as the tax code can change and it is difficult to assign a probability distribution to these changes.

6.2 Naïve diversification strategies

Discussions of tax risk in the context of retirement savings are often accompanied by a suggestion to diversify among saving vehicles. For example, in a Vanguard document, Ahern et al. (Reference Ahern, Americks, Dickson, Nestor and Utkus2005) state: ‘Pre-tax savings are more beneficial if a participant is in a lower tax bracket in retirement; Roth savings are more beneficial if a participant is in a higher bracket. In a world of uncertain future tax rates, participants should diversify. Just as they hold fixed income assets to diversify the risks of stocks, so participants should hold Roth savings to diversify the risks associated with pre-tax savings’. To investigate the potential benefits of having both types of accounts, we use the naïve diversification strategy defined in Section 2 where the individual allocates a constant proportion α of savings/withdrawals to the traditional account and a proportion 1 − α to the Roth account. Of course, this strategy is limited in the sense that the individual cannot adjust α in every period, but it gives us a simple platform to assess the welfare gains stemming from risk reduction. In the next subsection, we will discuss how mixed strategies can be improved over naïve ones.

Before evaluating the risk reduction benefits associated with a mixed strategy, we must first recognize that this approach can increase welfare even in the absence of risk. To understand why, consider the following example where the marginal tax rates before and after retirement are respectively 15% and 10%, but withdrawals in excess of $10,000 trigger the taxation of Social Security benefits and are subject to an effective marginal tax rate of 18.5%. An individual who has to make annual withdrawals of $15,000 can use a naïve strategy with α = 2/3 and withdraw $10,000 from a traditional account and $5,000 from a Roth account. This strategy is beneficial because it allows the individual to gain from the lower tax rate of 10%, while avoiding the higher rate of 18.5%.

For the case without tax risk, we compute the welfare gains associated with each α from 0 to 1 and thus are able to find the optimal α. The results are presented in the fourth and fifth columns of Table 2 for each of the cases considered previously. Figure 4 also gives a graphical representation of the welfare gains as a function of α for three representative cases. Those who do not encounter issues with the taxation of Social Security find it optimal to allocate 100% of savings to traditional accounts. A 100% Roth strategy is optimal for those whose effective marginal tax rates after retirement are higher than before retirement even before making any withdrawal. For those who are in the situation described in the previous paragraph (about 40% of our cases), it is optimal to divert some (but not all) savings to the Roth account to avoid the higher marginal tax rates. Note that the welfare gains with the optimal α are always non-negative: losses with traditional accounts can be avoided when there is an option to allocate part of savings to a Roth account. The value of the option to invest a fraction of savings in Roth accounts can be computed by taking the difference between the two welfare gains in Table 2: the last column of Table 2 shows that this value is on average $1,800 when positive, with a maximum of $4,518.Footnote ²²

Figure 4. Welfare gains with naïve diversification strategies (over Roth accounts) for selected cases (age 45 in 2010).

We now introduce tax variability with a simple no-drift scenario where marginal tax rates after retirement can go up or down by 20%. Using Section 2's notation, this translates into p ₁=p ₂ = 50%, θ₁ = 120%, and θ₂ = 80%. Figure 4 contrasts the welfare gains for the cases with risk (dashed lines) and without tax risk (solid lines). For those in the LHS/HS education groups, the lines coincide and tax risk has essentially no impact on welfare gains. For those in the college group, we observe a small difference and tax risk reduces welfare by at most $409 when α = 100%. The loss attributable to tax risk diminishes with diversification, for example it is cut by $281 when α = 50%. However, Figure 4 shows that risk reduction benefits are not the only consideration when choosing α: gains or losses in the scenario without risk must also be taken into account. In the previous example with α = 50%, diversification is suboptimal because the certain loss ($1,580) is much higher than the risk reduction benefits ($281).

More generally, in most of our illustrations the magnitude of the risk reduction benefit that a given α brings is much smaller than the corresponding gain or loss in the no-risk case. Indeed, the optimal proportions allocated to traditional accounts in Table 2 are essentially the same for the cases with and without tax risk. Although intuitive at first, the analogy with a diversified portfolio of stocks and bonds does not translate well because traditional and Roth accounts can have great differences in expected values, which dominate the volatility effect. On the other hand, the certainty case showed that the peculiar nature of the tax structure provides a new motivation for diversifying: a mix of traditional and Roth accounts can have a higher expected value than a linear combination of the pure cases.

6.3 Mixed strategies

The naïve diversification strategy suggested in the previous section can be fine-tuned to further improve welfare. For instance, instead of mixing both accounts in every period before retirement, the optimal strategy is likely to involve a switch between periods where contributions are either 100% traditional or 100% Roth.Footnote ²³ Unless there are major fluctuations in income, there should be relatively few transitions between the two accounts. Starting with the Roth account can be justified by lower marginal tax rates at that time or by concerns about early withdrawals and the 10% penalty tax.Footnote ²⁴ Conversely, starting with the traditional account is preferable when these are not an issue and marginal tax rates decline after retirement.Footnote ²⁵ If savings in the traditional account eventually reach a level such that the marginal tax rate after retirement exceeds the pre-retirement one, a permanent switch to Roth accounts would be recommended. Temporary moves to the Roth side could also be motivated in periods of income loss or with particularly high tax deductions. After retirement, the strategy would be a generalization of our earlier example with α = 2/3 to every period: withdrawals would be made from the traditional account first, and α would be chosen such that higher marginal tax rates are avoided.

7.1 Increases in retirement savings

Table 3 presents the gross and net levels of retirement savings at age 65 for each scenario, assuming that the individual was aged 45 in 2010. By education group, these range from: $40,000–$100,000 (LHS), $60,000–$140,000 (HS), and $50,000–$350,000 (college). Not surprisingly, higher income translates into higher savings, whereas higher pensions reduce the need to accumulate wealth. The bottom part of Table 3 tests the sensitivity of these results to the following changes in parameters: a = 25, a = 45, r = 1%, r = 5%, β = 1, β = 5, γ = 1, γ = 5, a reduction of 25% in Social Security benefits, and an increase by $5,000 in the exemptions amounts E and E_R. All cases are considered in the analysis, but due to space limitations the tables present only the averages for all education/income categories. The results show that the level of retirement savings is very sensitive to the choice of parameters, being halved or almost doubled in some scenarios.

Table 3. Retirement savings at age 65 (in dollars; age 45 in 2010)

The more salient result in Table 3 is the marginal impact of traditional accounts on retirement savings. First, those in the less-than-high-school category increase their savings by about $15,000. In their case, the gross and net differences are the same because they do not have enough income after retirement to pay taxes. The differences for the high-school group are much smaller: the gross increase is about $6,000 and the net increase averages only $200. Finally, the college group displays the more striking results. In their case, the gross increase can be pretty substantial reaching up to $68,000. However, once the increase in the tax liability is considered, this gain mostly vanishes with an average of only $2,800. Testing the sensitivity of these results, we find that the meaningful increase in net retirement savings for the less-than-high-school group is generally robust. The high-school group exhibits more variable results, ranging from −$20,000 to $20,000. The college group displays a pattern mostly similar to the one observed in the top portion of Table 3: a high increase in gross retirement savings is experienced, but most of it goes away once the net values are considered.

7.2 Income and substitution effects

Traditional accounts are more effective at increasing savings in some cases than others. To understand why, this section suggests a breakdown for the increase in retirement consumption (or equivalently the increase in net retirement savings) into an income effect and a substitution effect. The income effect comes from the additional tax subsidy that tax deductible contributions generate. It can be computed as the difference in the lifetime value of taxes as follows:

(25)$${\rm TS} \equals {\rm Tax}\;{\rm subsidy} \equals \int_{t_{\setnum{0}} }^{\omega } {{\rm e}^{ \minus r\lpar R \minus t\rpar } \lsqb {\rm tax}_{t}^{{\rm Roth}} \lpar s_{t} \rpar \minus {\rm tax}_{t}^{{\rm Trad}} \lpar s_{t} \rpar } \rsqb {\rm d}t.$$

It should be noted that retirement consumption does not increase by the entire amount of the tax subsidy, the wealth effect increases consumption in all periods, both before and after retirement.Footnote ²⁶ Introducing the notation $C_{t\comma T} \equals \int _{t}^{T} {\rm e}^{ \minus r\lpar R \minus s\rpar } c_{s}\ast {\rm d}s$, the proportion of the tax subsidy allocated to retirement consumption can be estimated with the following ratio:

(26)$$q \equals {\rm Propensity}\;{\rm to}\;{\rm consume}\;{\rm after}\;{\rm retirement} \equals {{C_{R\comma \omega } } \over {C_{t_{\setnum{0}} \comma \omega } }}.$$

A reduction in tax rates after retirement not only creates a subsidy but also a substitution effect by lowering the relative price of post-retirement consumption. The value of the new savings before retirement associated with the substitution effect can be computed with ${\rm New}\;{\rm Savings} \equals C_{t_{\setnum{0}} \comma \omega }^{{\rm Roth}} \lpar q^{{\rm Trad}} \minus q^{{\rm Roth}} \rpar $. Accordingly, the total changes in pre- and post-retirement consumptionFootnote ²⁷ can be written as

(27)$$\eqalign{ \tab {\rm Change}\;{\rm in}\;{\rm pre {\hbox {-}} retirement}\;{\rm consumption} \equals \lpar 1 \minus q^{{\rm Trad}} \rpar \cdot TS \minus {\rm New}\;{\rm Savings} \cr \tab \quad {\rm Change}\;{\rm in}\;{\rm retirement}\;{\rm consumption} \equals q^{{\rm Trad}} \cdot TS \plus {\rm New}\;{\rm Savings} \cr} $$

Table 4 presents the results of the decomposition for the change in retirement consumption along with the tax subsidies and the applicable effective marginal tax rates. The table includes the previous cases from Figure 3 and we will use them as representative examples. The average tax subsidy in Table 4 is about $10,000; since q ^Trad is generally around 20%, the average increase in retirement consumption attributable to the tax subsidy is only $2,000. By themselves, subsidies can be a relatively expensive way to generate a small increase in retirement consumption. For example, the increase is at best $9,240 in Case 6, but the total cost of the subsidy is $48,396.

Table 4. Increase in retirement consumption

The substitution effect can generate larger increases in retirement consumption than the income effect, but only for some groups. It works well for those who pay little or no taxes after retirement, for example in Cases 1 and 2 the associated increases in retirement consumption are $13,745 and $14,248. For those who pay meaningful taxes after retirement, the taxation of Social Security benefits again changes the cards and reduces the substitution effect's potential. In many cases, marginal tax rates increase and the substitution effect is actually negative. For example, in Case 5 the marginal tax rate after retirement jumps to 27.75% and retirement consumption is reduced by $13,214.

To conclude this section, it is interesting to put the tax subsidies in perspective by observing that they move in sync with the welfare gains in Table 2. In other words, welfare gains associated with the tax deductibility of contributions arise because they are paid for by tax subsidies. Tax deductible contributions do not generate a benefit above their cost: on average, welfare gains are $500 less than tax subsidies because consumption patterns are disrupted. Although tax deductible contributions benefit most people in partial equilibrium, it should be kept in mind that this is not necessarily the case in general equilibrium where tax subsidies have to be financed by an increase in other taxes.