Social Security: calculating a robust retirement age

Introduction

First of all, I’m not an economist. This exercise is just something I did out o curiosity. I wanted to find out if there’s such a thing as a robust retirement age that would cover a relatively large combination of economic conditions: aging population, longer life expectancy, variation in the contributions to Social Security, changes to the population, legal working age, etc.

The math I used, in my opinion, models the Social Security system in a very simplistic way: I do not account for inflation and I do not model any particular function for changes in the working population.

Nevertheless I thought I got some interesting results and didn’t see any harm in sharing them here.

The Model

I’m doing my calculations based on the current model for Social Security where the active workforce pays for the retired individuals. So the simple model would be something like this:

As you can see it’s a very simplistic model.

The difference between the retirement age and the legal working age multiplied by the number of active works per year is the total number of active workers; similarly, the difference between the average life expectancy and the age of retirement multiplied by the number of retirees per year is the total number of retirees.

We are interested in calculating the retirement age based on all the other variables so after some algebra we get:

You can see this equation as nothing more than the weighted average between the legal age of employment and the life expectancy.

We can simplify the equation a little further by postulating that the average number of active workers per year is proportional to the average number of retirees per year, something like this:

Suppose that for 10 years the average number of new employees is 100 per year; then after 10 years falls to 90, then 80, then 70; let’s also assume for this example that workers retire after 30 years of labor.

Under these constraints, after 40 years, we have:

1000 retirees over a span of 10 years = average 100 retirees per year
2400 active workers over a span of 30 years = average 80 active workers per year

So we can say the number of active workers per year is 80% of the current number of retirees per year.

In real life this number would be much more complex to calculate but it servers our simple model well.

Making the two values proportional by a value b we have:

This result is somewhat intuitive: one would expect the longer life expectancy would pull the retirement age up; if the salary contribution a is too low and the number of retirees too high when compared to the number of active workers the retirement age goes up.

It’s also possible to see that because the product of a to the legal working age is a small value a modest increase in a (contribution to Social Security) can significantly reduce the retirement age since the numerator will not have an as significant increase as the denominator.

But I digress. What I wanted to find out is if there’s such thing as a robust retirement age. For that, I used the good old Monte Carlo simulation.

Method

The idea behind using Monte Carlo simulation for this exercise is to calculate the retirement age for random combinations of values of almost all variables involved — within a certain finite range — and then build an histogram of the resulting ages; if I get the classes for the histogram right I should get a relatively nice distribution that can be analyzed.

We can control 3 variables in the equation: 1) the percentage of the average salary of active workers — c — that is paid as retirement; 2) the percentage of salary from active workers —a — contributed to Social Security; 3) the legal age of employment.

I decided to study 5 cases, one for a different value for c: 20%, 40%, 60% 80% and 100%. The remaining variables will be randomly distributed in the same arbitrary intervals for all cases.

I’ll arbitrarily select the robust age for retirement as the one that covers 80% of the distribution generated by 11,000+ random samples. I used Calc from LibreOffice for the calculations.

Why 80%?

I accept every time I get in my car there’s 20% chance I could die — Niki Lauda

If it’s good for Niki Lauda, it’s good for me.

So here are the ranges I used for each variable in all cases:

The average life expectancy is predicted to only go up by one year by 2040). The percentage of salary paid to Social Security today is 12.50% (6.25% paid by the worker and 6.25% paid by the company); I added quite extreme cases for the range of this variable just to see how it behaves.

Similarly, I tried to use relatively large ration for the ratio between active employers and retirees.

Case 1: c = 20%

The 80% limit falls in the range below 68 years-old. This is very interesting, considering that today the Social Security pays about 25% of the active worker salary and the retirement age is set to be 67 years-old in a few years; this retirement age seems to be a robust choice — based on this model — for a large percentage of different combinations of economic scenarios.

Case 2: c = 40%

Doubling the amount paid by social security shifts the histogram to the right, which is not surprising. For this case, the age of retirement in the 80% case is 72 years old; not a very attractive prospect.

Calculating for the other cases didn’t give much additional information, the retirement age went up to the 74–76 range, virtually meaning no one retires.

Conclusion

In Economy it’s not rare finding simplified mathematical models for complex economic processes. One that comes to mind is the concept of velocity of money, which tries to relate price with the demand for money (see https://en.wikipedia.org/wiki/Velocity_of_money) and mathematically predict human behavior.

Economic processes — especially in free market economies — are difficult to model because it’s a human process. It is heavily affected by the confidence that people have in the market and its many parts.

Nevertheless, this exercise — at least for me — is a good example that even a simplistic model can be used, along with suitable statistical tools, to make reasonable economic decisions. Monte Carlo simulations are used frequently in risk analysis following a methodology more or less similar to the one I applied here (see https://www.investopedia.com/terms/m/montecarlosimulation.asp).