Dynamic analysis of the effect of immigration on the demographic background of the pay-as-you-go pension system Massimo Angrisani Sapienza University of Rome, Italy Anna Attias Sapienza University of Rome, Italy Sergio Bianchi University of Cassino, Italy Zoltàn Varga Szent Istvàn University, Hungary MAF 2010, Ravello April 7th-9th 2010
26
Embed
Dynamic analysis of the effect of immigration on the ... · PDF fileDynamic analysis of the effect of immigration on the demographic background of the pay-as-you-go pension system
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Dynamic analysis of the effect of immigration on the demographic background
of the pay-as-you-go pension system
Massimo Angrisani Sapienza University of Rome, Italy
Anna Attias
Sapienza University of Rome, Italy
Sergio Bianchi University of Cassino, Italy
Zoltàn Varga
Szent Istvàn University, Hungary
MAF 2010, Ravello April 7th-9th 2010
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
MAF 2010, Ravello April 7th-9th 2010
Outline• Problem setting and motivation
• The model:• The Leslie model• The modified model• A stabilization theorem• Adding immigration• Controlling the population to demographic equilibrium
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
MAF 2010, Ravello April 7th-9th 2010
The modified population model
Let• N be the maximal age for each sex;• xF(t) be the female population vector;• xM(t) be the male population vector;• x(t) be the whole population vector;• fi
be the per mil fertility;• qi
F
be the female mortality rate;• i
F
= 1 –
qiF
be the female survival rate;
• be the sex ratio (at birth)
The Leslie model InsectsOur modified Leslie model Human population
0 0
0 0 0
( ) ( )( ) ( ) ( )
F F
F M
x T x Tx T x T x T
The female per capita birth rate is (i=15,…,50)1000
Controlling the population to demographic equilibrium
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
Theorem C
[Angrisani, Attias, Bianchi, Varga (2010)]The age distribution of population converges to the equilibrium age distribution, i.e.
( 1)limt
x t zu
Remark 2. Theorems B and C hold if ( t) is replaced by any positive sequence tending to zero. This choice makes the application of the model more flexible.
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
MAF 2010, Ravello April 7th-8th 2010
•
The control model is equipped with a parameter of convergence (between 0 and 1) that regulates the speed of convergence and the total immigration at the same time. A value near 1 slows down the convergence but limits the yearly admission of immigrants.
•
In our simulations this parameter is set to 0.9; and the algorithm was modified, admitting only immigrants under age 35. This modification doesn’t change the convergence of the algorithm.
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
MAF 2010, Ravello April 7th-8th 2010
Figure 1. Inverse old-age dependency ratio, plotted against time (in years)
Figure 2. Total population size, plotted against time (in years)
Scenario (A): constant yearly immigration –
180.000
Relatore
Note di presentazione
For all the figures the starting time 0 is the year 2006. The simulation concerns one hundred years ahead. Scenario A The inverse old-age dependency ratio (Figure 1) declines from 3.2 to 1.6 (minimum which reaches in 43 years) and then shows an upward movement which takes it to about 2.15 in one hundred years. The dramatic fall of the ratio corresponds (Figure 2) to the decline of the population, which passes from about 58,5 millions to 52 mllions (at the minimum ratio time) and then continues falling to about 40 million people.
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
MAF 2010, Ravello April 7th-8th 2010
Figure 3. Inverse old-age dependency ratio, plotted against time (in years)
Figure 4. Total population size, plotted against time (in years)
Scenario (B): constant yearly immigration –
494.289
Relatore
Note di presentazione
Scenario B The inverse old age dependency ratio (Figure 3) declines from 3.2 to about 1.92 (minimum which reaches 40 three years) and then shows an upward movement which takes it to about 2.6 in one hundred years. As a consequence of the larger number of immigrants allowed (494.289), the ratio declines at a slower rate with respect to the previous case but the positive trend of the ratio itself is balanced by (Figure 4) an excess of population that reaches about 69 millions at 40 years ahead and reaches about 87 millions at 100 years ahead.
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
MAF 2010, Ravello April 7th-8th 2010
Figure 5. Inverse old-age dependency ratio, plotted against time (in years)
Figure 6. Total population size, plotted against time (in years)
Scenario (C): Control model, initial immigration = 450.461
Relatore
Note di presentazione
Scenario C The inverse old age dependency ratio (Figure 5) shows the best performance among the three cases here considered. Its decline is non monotonic and the minimum is reached at about 60 years ahead (about 2.2 against 1.92 and 1.6 of the previous scenarios). A slow growth brings the ratio to the value of 2.3 on the 100 years horizon. It is worthwhile observing that the in this case population size (Figure 6) remains acceptable: it grows from about 58,5 millions to 68,5 millions up to the first thirty years and then slowly declines back to 58 millions on a one hundred years horizon.
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
MAF 2010, Ravello April 7th-8th 2010
Figure 7. Total population size,
for scenario C), plotted against time (years)control model, initial
immigration= 450.461, time horizon= 200 years
Relatore
Note di presentazione
Figure 7 shows the long-term stabilization of the total population size for Scenario C.
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
MAF 2010, Ravello April 7th-8th 2010
Figure 8. Yearly total immigration,
for scenario C), showing long-term stabilization of immmigration policy, initial
immigration= 450.461,
Relatore
Note di presentazione
On Figure 8, a quick stabilization tendency can be seen, following a first pick, from year 20 on, we can observe that the algorithm allows an immigration oscillating around relatively low level.
Angrisani, Attias, Bianchi, Varga, “Immigration and demographic equilibrium”
MAF 2010, Ravello April 7th-8th 2010
Discussion
Classical Leslie population growth model can be extended to a control-theoretical model,
appropriate for dynamic simulation of
the demographic background of a pension system.
In the control model, the yearly immigration can be determined for each age class, such that
the age distribution of the population moves towards a demographic equilibrium,
each year the total immigration is minimized,
a parameter can be set to regulate the speed of convergence and the total immigration at the same time.
The efficiency of the algorithm was shown, by comparing it to constant immigration scenarios, in terms of the active/pensioner