default on plan enrollment differs by gender to investigate whether females are more susceptible to the default and we fi nd no difference.
6
In summary, we fi nd a substantial effect of the default on plan choice: The default increases the probability of enrolling in one plan over the other by 60 percentage
points. This effect is, to our knowledge, the fi rst estimate of causal effect of the default in the context of the choice between DB and DC plans. There are several potential
mechanisms behind the effect of the default we observe, including inertia, transac- tion costs, complexity, inattention, anchoring, procrastination, or the perception of
endorsement Beshears et al. 2008; Bernheim, Fradkin, and Popov 2011. While the magnitude of the estimated effect is comparable to prior work on defaults in the retire-
ment savings literature, the effect estimated here is somewhat surprising given the potentially large differences in wealth accrual across the two plans and the irrevers-
ible nature of the decision Beshears et al. 2008. In addition, the heterogeneity in the default across age provides a unique source of identifi cation for estimating the causal
effect of the default.
IV. Solving for the Optimal Age- Based Default Rule
The estimates of the default shown in the previous section along with those found in prior literature indicate that defaults can be powerful tools to steer
economic behavior. An important question that has been broached in the literature on defaults and in policy circles is how defaults may be constructed to improve welfare.
In this section we explore: 1 how to set an optimal default based on observable characteristics namely age; and 2 how to determine the value of incorporating het-
erogeneity into default provisions based on age relative to alternative default policies.
7
A. Characterizing retirement wealth in the DB and DC plan
We begin by characterizing the future benefi ts that accrue in both the DB and the DC plan.
8
The DB formula is defi ned by b
j
w
j
for all years j between 0 and r- a, where r is the age of exit from the fi rm and a is the worker’s current age, as a function of annual
wages in year j, w
j
. The function b
j
is completely general in that it can allow for vari- ous types of accrual patterns.
9
The wealth evaluated at retirement age ρ in the DB plan
as a function of age a is then:
6. Results are available upon request. 7. Note that the approach outlined in this section need not be limited to the particular fi rm in this paper; in
theory, any fi rm’s employee and pension plan characteristics could be utilized to obtain the optimal age- based default rule for that fi rm.
8. At the fi rm in our setting, while the employee had also accrued years of service prior to the plan migration, the employee received that benefi t stream regardless of the plan they were enrolled in after the transition.
Therefore, the value of past benefi ts accrued are not included in wealth calculations for either the DB or DC plan and are not relevant for the comparison of future retirement wealth across the two types of plans
9. or example, a fi rm that offers 2 percent of average wages for each year of service as the annual retirement benefi t would have b
j
w
j
= 0.02w
j
for all j. Similarly, a fi rm that offers its employees 2 percent × years of service × the average salary over the last fi ve years at the fi rm as the annual retirement benefi t would be
characterized by b
j
w
j
= 0 if j r – a – 5 and b
j
w
j
= 0.02r – aw
j
5 if j ≥ r – a – 5.
4 w
DB
a =
r −a
∫
b
j
w
j
A
ρ
dj, where
A
ρ
is the actuarial present value of a stream of 1 annual payments commencing at age
ρ and paid until death, which therefore differs by gender due to differences in life expectancies.
The wealth in the DC plan at retirement age ρ is equal to the contributions made,
accumulated with returns from investment experience. Contributions are typically a percentage of annual wages, though they can vary from year to year. We denote em-
ployer contributions into the employee’s account in year j by c
j
and the sequence of returns in all subsequent years by k for k in [ j,
ρ – a].
10
The wealth evaluated at retirement age
ρ in the DC plan as a function of age a is then: 5
w
DC
a =
r −a
∫
c
j
w
j
e
∫ j
ρ−aδkdk
dj. We next compute the expected discounted utility of retirement wealth for each plan
by explicitly modeling two sources of uncertainty: separation risk, which affects the age of exit r, and investment risk, which affects the sequence of returns ∙.
11
We assume r and ∙ are drawn from a joint distribution
h
p
r, δ | a r ≤ r for plan p in
{DB, DC}. Separation risk affects the value of both the DB and the DC plan: If the
employee leaves the fi rm, both contributions to the DC account and benefi t accruals in the DB plan cease. The degree to which separation risk affects the relative values of
the two plans depends on whether the rate of wealth accrual is front- ended, back- ended, or age- neutral. As DB wealth accrual is typically more concentrated toward the
end of one’s career, the risk that an employee leaves the fi rm may affect DB wealth accrual more than DC wealth accrual. On the other hand, investment risk only affects
DC retirement wealth as DB wealth does not depend on ∙.
12,13
Assuming a discount rate , the expected utilities for each plan are given by: 6
EU w
DB
a =
−∞ ∞
∫
a r
∫
U w
DB
a 1 + β
ρ−a
h
DB
r, δ|a r ≤ r drdδ
7 EU w
DC
a =
−∞ ∞
∫
a r
∫
w
DC
a 1 + β
ρ−a
h
DC
r, δ|a r ≤ r drdδ
We then defi ne the certainty equivalent for plan p in {DB, DC}
as:
10. Note we assume c
j
includes only employer contributions, not employee contributions. This distinction is made so both plans’ values refl ect the benefi ts provided by the employer only.
11. We do not explicitly model other sources of risk, such as the risk that the employer goes bankrupt and can no longer honor DB pension obligations, or the risk of dying prior to retirement. However, it can easily
be shown that the results presented here are general to any types of risk. 12. In our numerical results that follow, we assume independence between separation risk and investment
risk and that the distribution does not vary by plan choice. We assume r is drawn from a distribution fr|a r
≤ r¯ = [ fr] [1 – Fa] where F is the cumulative distribution function of f and that the sequence ∙ is determined by a draw from a distribution g . However, these assumptions do not affect our analytical
results, which are all in terms of the certainty equivalent. 13. Our analysis omits future opportunities for pension wealth accrual in other jobs after separation. If there
is signifi cant backloading of DB wealth accrual, this omission will tend to undervalue the DB plan relative to the DC plan in our framework.
8 CE
p
a = U
−1
EU w
p
a. The certainty equivalent
CE
p
a for plan p is the amount that makes the individual
indifferent between receiving the amount CE
p
a for certain and the gamble character-
ized by the uncertain income stream from plan p. Therefore, plan p is preferable to
plan ˆp if and only if
CE
p
a CE
ˆp
a.
B. Solving for the Optimal Default Rule
