Table VII.

Robustness

	Specification	Average wealth equivalent	Average absolute welfare difference (million pounds)
			Symm. info.	Mandate 0	Mandate 5	Mandate 10
1	Baseline specification	100.16	126.5	−107.3	2.1	126.7
	Different choices of γ's:
2	Consumption γ=5, Wealth after death γ=5	100.51	111.0	−117.0	0.0	111.0
3	Consumption γ=1.5, Wealth after death γ=1.5	99.92	133.2	−102.0	0.6	133.2
4	Consumption γ=3, Wealth after death γ=5	100.47	120.0	−123.0	3.0	120.0
5	Consumption γ=3, Wealth after death γ=1.5	99.94	135.3	−96.9	2.1	135.3
6	Row 5 + allow heterogeneity in initial wealtha	101.18	127.4	−148.3	−32.9	128.8
	Other parameter choices:
7	r=0.05 and δ=0.05	99.29	119.4	−97.5	5.7	119.4
8	January 1990 annuity rates	100.16	123.0	−112.5	0.0	123.0
	Wealth portfolio outside of compulsory annuity:
9	Fraction annuitized (η) = 0.3	100.65	114.0	−118.0	0.0	114.0
10	Fraction annuitized (η) = 0.1	99.93	135.0	−108.0	−4.2	135.0
11	Allow heteregoeneity in ηb	100.22	141.3	−113.7	2.5	132.4
12	Half of initial wealth in public annuityc	99.95	255.6	−426.3	−34.2	243.6
	Parametereization of heterogeneity:
13	Non-Gompertz mortality distributiond	100.06	144.0	−100.8	6.0	144.0
14	α dist. Gamma, β dist. Lognormal	100.20	132.0	−111.6	3.0	132.0
15	α dist. Gamma, β dist. Gamma	100.14	123.0	−105.6	3.0	123.0
16	Allow covariatese	100.17	132.0	−110.1	3.0	132.0
17	β fixed, Consumption γ heterogeneousf	100.55	129.3	−110.0	2.1	129.4
18	Heterogeneity in both β and γ	100.05	131.9	−117.0	−5.9	129.0
	Different information structure
19	Biased beliefs: θ = 0.5	100.16	122.9	−104.0	3.0	122.9
20	Biased beliefs: θ = 2	100.19	126.0	−101.6	5.9	126.0
21	Uncertain α: σε = 0.027	100.15	128.9	−104.7	5.9	128.9
22	Uncertain α: σε = 0.108	100.17	126.0	−105.9	3.0	126.0
	Departure from neo-classical model:
23	Some individuals always “pick the middle”g	100.22	132.0	−99.9	9.0	132.0
	Different sample:
24	“External” individualsh	95.40	137.4	−134.4	−16.8	137.7

The table reports summary results – average wealth equivalent and average welfare effects – from a variety of specifications of the model. Each specification is discussed in the text in more detail. Each specification is shown on a separate row of Table VII and differs from the baseline specification of Table VI (which is reproduced in the first row of Table VII) in only one dimension, keeping all other assumptions as in the baseline case.

^aSee text for the parameterization of the unobserved wealth distribution. For comparability, the average wealth-equivalent is normalized to be out of 100 so that it is on the same scale as in the other specifications.

^bSee text for the parameterization of the unobserved fraction of non-annuitized wealth (η) distribution.

^cWe assume the public annuity is constant, nominal, and actuarially fair for each person.

^dThis specification uses hazard rate of ${\alpha }_i\exp \left(\lambda {(t-t_0)}^h\right)$ with h = 1.5 (Gompertz, as in the baseline, has h = 1).

^eCovariates (for the mean of both α and β) consist of the annuitized amount and the education level at the individual's ward.

^fβ is fixed at the estimated μ_β (see Table III). Since the resulting utility function is non-homothetic, we use the average wealth in the population and renormalize, as in row 6. See text for more details.

^gThe welfare estimates from this specification only compute welfare for the “rational” individuals, ignoring the individuals who are assumed to always pick the middle.

^h“External” individuals are individuals who did not accumulated their annuitized funds with the company whose data we analyze. These individuals are not used in the baseline analysis (see Appendix B).