Risks linked to an imbalanced pension scheme with a changing population

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death.

For instance, , where t is the year.

is the infantile mortality rate, which is 4 ‰ (INSEE’s data) ([9]).

We took and from INED’s statistics ([11]).

We suppose = 1/20, since only 1/20 of young people grow to laborers. In fact, we can adopt this estimation because the mortality rate in the period 2-19 years old is almost a constant (you can consult INED’s statistics [11] to verify).

The same reasoning is not possible to estimate as 1/30 because the mortality rate is not constant (it rises from 0.6 to 3.6), even if we are obliged to make a weighted average to fix in the model.

Therefore was estimated so that we find the same figures as INSEE’s statistics in [5]’s Table 4 (0.021≈1/47)

The inputs of the model are the number of births N (taken from INSEE’s statistics [7] in 2007 and 2008 then kept constant), the inflation rate ([6]), the average pension b and the average wage w ([8]). The initial data regarding the population are from 2007. Table 4 of [5] shows the population distributions by age.

We set the pension that the State gives to the pensioners. In 2008, the State gave a total of 261G€ ([4]), which represents 18.54K€ per pensioner. For the following years and for 2007, the pensions are indexed on the inflation rate.

The contribution rate q is the output variable of the model, which means that the risk is totally borne by the working generation if the population changes.

First, we can compute the total amount of pensions with (2.3), then we can compute q with (2.5).

With Excel, we obtain the following table:

Table 1: Simulation 1 – Basic model

t Inflation b (K€) q w (K€) N (M) Y (M) L (M) P (M) Population (M) Total pension (G€) Total contribution (G€)

2007 1,5% 18,27 20,93% 34,49 0,819 15,92 34,17 13,502 63,60 247 247

2008 2,8% 18,54 22,17% 34,49 0,828 15,95 34,14 14,076 64,16 261 261

2009 0,1% 18,56 23,10% 34,49 0,828 15,97 34,10 14,64 64,72 272 272

2010 1,0% 18,74 24,25% 34,49 0,828 15,99 34,07 15,20 65,27 285 285

2011 1,9% 19,10 25,63% 34,49 0,828 16,01 34,05 15,75 65,81 301 301

2012 2,2% 19,52 27,12% 34,49 0,828 16,03 34,02 16,30 66,35 318 318

2013 1,0% 19,72 28,32% 34,49 0,828 16,05 33,99 16,84 66,88 332 332

2014 0,8% 19,87 29,47% 34,49 0,828 16,07 33,97 17,37 67,41 345 345

2015 0,0% 19,87 30,38% 34,49 0,828 16,08 33,95 17,90 67,93 356 356

2016 2,0% 20,27 31,91% 34,49 0,828 16,10 33,93 18,42 68,45 373 373

We observe that q increases without any change in the population, which means that the system is unstable by its nature.

Now, we are going to see the consequences of the increase in life expectancy.

Instead of taking 0.0057, we take decreasing values.

Table 2: Simulation 2 – Effect of an increasing life expectancy

t Mortality Inflation b (K€) q w (K€) N (M) Y (M) L (M) P (M) Population (M) Total pension (G€) Total contrib. (G€)

2011 0,0057 1,9% 19,10 25,63% 34,49 0,83 16,01 34,05 15,76 65,81 301 301

2012 0,0055 2,2% 19,52 27,12% 34,49 0,83 16,03 34,02 16,30 66,35 318 318

2013 0,0053 1,0% 19,72 28,32% 34,49 0,83 16,05 33,99 16,84 66,89 332 332

2014 0,0051 0,8% 19,87 29,48% 34,49 0,83 16,07 33,97 17,38 67,41 345 345

2015 0,0049 0,0% 19,87 30,41% 34,49 0,83 16,08 33,95 17,91 67,94 356 356

2016 0,0047 2,0% 20,27 31,96% 34,49 0,83 16,10 33,93 18,43 68,46 374 374

2017 0,0045 2,0% 20,68 33,55% 34,49 0,83 16,12 33,91 18,95 68,98 392 392

2018 0,0043 2,0% 21,09 35,20% 34,49 0,83 16,13 33,89 19,47 69,49 411 411

We observe greater variations on q than those found in the previous simulation.

The variation from the first simulation is small since it is a long-term modification. However, we cannot forecast 20 years forward with this model since the parameters will probably change over time.

4. SOLUTIONS

4.1. Political solutions

An easy political solution is to increase the period of contribution. We tested this solution assuming that the reformed system started in 2011, which was the case in France. INSEE’s statistics [10] show that 26% of the population above 60 years old is under 60 and 64 years old. That’s why we chose that 26% of the pensioners would keep on working and contributing.

Table 3: Simulation 3 – Effects of the reform

t Inflation b (K€) q w (K€) N (M) Y (M) L (M) P (M) Population (M) Total pension (G€) Total contribution (G€)

2007 1,5% 18,27 20,93% 34,49 0,82 15,9 34,17 13,5 63,60 247 247

2008 2,8% 18,54 22,17% 34,49 0,83 15,9 34,14 14,1 64,16 261 261

2009 0,1% 18,56 23,10% 34,49 0,83 16,0 34,10 14,6 64,72 272 272

2010 1,0% 18,74 24,25% 34,49 0,83 16,0 34,07 15,2 65,27 285 285

2011 1,9% 19,10 16,93% 34,49 0,83 16,0 34,05 15,8 65,81 223 223

2012 2,2% 19,52 17,85% 34,49 0,83 16,0 34,02 16,3 66,35 235 235

2013 1,0% 19,72 18,56% 34,49 0,83 16,1 34,00 16,8 66,89 246 246

2014 0,8% 19,87 19,25% 34,49 0,83 16,1 33,97 17,4 67,41 255 255

2015 0,0% 19,87 19,77% 34,49 0,83 16,1 33,95 17,9 67,93 263 263

2016 2,0% 20,27 20,69% 34,49 0,83 16,1 33,93 18,4 68,45 276 276

2017 2,0% 20,68 21,63% 34,49 0,83 16,1 33,91 18,9 68,96 290 290