Dissertation

Figure 3.2: Baseline cross-sectional age-consumption profiles by efficiency level.

50

1

45

40

35

30

25

20

15

10

0

5
2

3

4

5

Student Version of MATLAB

25 40 55 70 85 100

Age

Labor hours per week

Consumption
Figure 3.3: Baseline cross-sectional age-labor hour profiles by efficiency level.

wealth equals zero. Mathematically, it is a solution to the following equation


52

Z0_T¯

exp {(g − β(ϕ))s} Q(s)θ {1 − l(s; ϕ)} w(t)ϕe(s) ds

T_r
= ^Z
_T^¯
exp {(g − β(ϕ))s} Q(s)b(t; ϕ) ds (3.63)

where s is model age. Note that the first term in the above equation gives the present

value of total benefits received and the second term gives the present value of the total tax

payments. Numerically solving equation (3.63) for each efficiency group yields the follow-

ing IRR distribution under the baseline calibration: {0.0479, 0.0323, 0.0256, 0.0218, 0.0192}.

Comparing these IRRs with the equilibrium rate of return from capital stock shows that

the bottom two quintiles of the model population experience a welfare gain from partici-

pating in the social security program, and the top three quintiles experience a welfare loss.

With ζ(ω) = 0.46, the social security program partially insures the households against un-

favorable efficiency realizations: the benefit annuity of the poorest households is close to

half of that of the wealthiest, when the labor income of the former is only one-fifth of the

latter (as ω = 0.2). The gross benefit replacement rates in the model population under

the baseline calibration are compared to the ones implied by the U.S. PIA benefits for- mula in Figure 3.4.⁹ The optimal retirement age distribution in the baseline calibration is

{61.48, 63.09, 63.61, 63.86, 64.02}, which shows that households with lower efficiency retire

earlier than those with higher efficiency.
3.6 Population aging

Population aging in the current model is driven by an increasing life expectancy and a

falling population growth rate. The 2009 OASDI Trustees Report published by the SSA

contains the long-range social security area population and the average life expectancy at ⁹_{It may be noted here that even though social security contributions and benefits are capped in the real}

world, no such cap exists in the model. However, the baseline distribution of income in the model is such that these caps would not bind even if they existed. As Huggett and Ventura (1999) point out, the maximum creditable earnings in the U.S. social security program averaged at 2.47 times the average earnings over the period 1990-94. The ratio of maximum earnings to average earnings in the baseline calibration is 1/0.6=1.67, which is lesser than 2.47.

1

0.9

0.8

0.7

0.6

0.5

0.4

U.S. Model

53

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Multiple of average earnings