Dynamic analysis of the effect of immigration on the ... · PDF fileDynamic analysis of the...

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”


Outline• Problem setting and motivation

• The model:• The Leslie model• The modified model• A stabilization theorem• Adding immigration• Controlling the population to demographic equilibrium

• Immigration scenarios and simulations

• Discussion

• References

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Note di presentazione

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Problem Setting and motivationDemographic equilibrium as a key variable

to ensure the

sustanaibility of a pay-as-you-go pension system.

“Ideal”

shape

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Problem Setting and motivationThe problem is that population pyramids of mature economies often display “critical”

features

“Bad”

shape

“Few”

contributors…

…for too many pensioneers

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Problem Setting and motivation

2007 -

Starting year 2051 –

Mid scenario

2051 –

High scenario 2051 –

Low scenario

age

age

age

age

Italian: M Foreigner: M

F F

With respect to previous estimates, ISTAT forecasts that the Italian population will grow more than 6 million people as a consequence of immigration.

In 2051 immigrants will represent 16,1%-18,4% of the whole population.

The basic idea:

Immigrants as a resource for stabilizing the population distribution in order to achieve the sustainability of the pay-

as-you-go pension system.

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Problem Setting and motivationProsRejuvenating the age structure of the population owing to two main reasons:

• Immigrants are generally young (immediate effect)• Immigrants generally display ferility rates higher than the

Italian one (postponed effect)

2007 2051Italian 1,36 1,39

Immigrants 2,35 1,86

ISTAT Fertility rates (mid scenario)

Convergence to the reproductive behaviour of the Italian females

RemarkThe analysis of the demography for the pension system is the very first step towards a more general solution also including the economic

component.

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The basic population modelConsider a population without sex structure where:- N

N is the upper bound of the age of an individual;

- xi

(t) is the number of individuals of age belonging to [i, i +1[ (for i = 0, 1,…, N -1) at time t;

- i

≥

0 average per capita birth rate in the i -th age group;- 0 < i

< 1 is the survival rate from age group i to i +1;

0 1 2 1

0

1

2

...0 ... 0 0

0 0 0. . ... . .0 0 0

N N

N

L

0 1 1( ) ( ), ( ), ..., ( ) TNx t x t x t x tThe population vector results

and the system matrix is

The population dynamics (so called Leslie model) is ( 1) ( )x t L x t

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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

F ii

f

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The modified population modelThe modified Leslie female matrix becomes

15 50

0

1

2

0 0 ... ... 0 ... 0 00 ... 0 ... 0 0 ... 0 0

0 ... 0 ... 0 0 ... 0 0. . ... . ... . . ... . .0 0 ... 0 ... 0 0 0 0

F F

F

F F

FN

L

The dynamics of females reads as ( 1) ( )F F Fx t L x t

The modified Leslie male matrix becomes

0

1

2

0 0 ... 0 ... 0 0 ... 0 00 ... 0 ... 0 0 ... 0 0

0 ... 0 ... 0 0 ... 0 0. . ... . ... . . ... . .0 0 ... 0 ... 0 0 0 0

M

M M

MN

L

The dynamics of males reads as 1(1 )( 1) ( ) ( )M M M F Fx t L x t e L x t

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A stabilization theoremSetting and denoted by

the block matrix synthesizing the whole population (L1M

is the (N-1)N matrix obtained by cutting the first row of LM), the dynamics is

1

0

F

M

L OdO L

15 151 10 ... 0 ... 0 ... 0F Fd

( 1) ( )x t x t

ProblemThe matrix

is neither irreducible nor primitive and hence the Perron-

Frobenius conditions ensuring the existence of a dominant positive eigenvalue are not satisfied.

SolutionWe prove that sub-matrix is irreducible

and primitive. Applying Perron-Frobenius to L50F

we get the asymptotical distribution of the sub-population and prove that it can be extended to the whole population.

15 49 50

0

50 1

49

0 0 ... ...0 ... 0 ... 0 0

0 ... 0 ... 0 0... ...

0 0 ... 0 ... 0

F F F

F

F F

F

L

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Theorem A (2006)There exists a population state x0

, which is a nonnegative eigenvector of the matrix , associated with a positive eigenvalue 0

. On the long term, the population

age distribution tends to the equilibrium age distribution in the

sense that, for any initial state x(0), there exists s > 0 such that

0

0

( )lim tt

x t sx

We state the following

A stabilization theorem


.

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Adding immigrationPurposes• Equipping the model with immigration

•

Setting up a dynamic model in order to steer the population by optimizing the immigration flow with respect to the pension system requirements

Denoted by:• i

’

the immigrant female reproduction rate• yi

F(t) [yiM(t)]

the immigrant female [male] population vector

one has


• the female reproduction rate dynamics( ) ( )( 1) : ( )

( ) ( ) ( ) ( )

F F Fi i i

i i iF F F F F Fi i i i i i

x t y tt tx t y t x t y t

(0) : Fi i i (i = 15,…, 50)• Basic assumption:

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Adding immigration


• the matrix

1

( )( ) : ( ) 0

F

M

L t Ot d t

O L

where the blocks depend on the new reproduction rates;

1(1 )( 1) ( ) ( ) ( ) ( )M F M F F Mx t L t x t e L x t y t

( 1) ( ) ( ) ( )F F F Fx t L t x t y t

• the dynamics of population

or, in terms of the whole population, setting y = [yF

yM]T

( 1) ( ) ( ) ( )x t t x t y t

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Controlling the population to demographic equilibrium


Purpose

(1) : (0) (0) (0)x x y Finding an immigration vector y(0) s.t. the new population state

provides an equilibrium age distribution for system matrix

.

Since, by Theorem A, there exists a normalized positive eigenvector z(0), corresponding to the positive eigenvalue 0

of (0), we have to minimize in u > 0

(0)(0) ( (0) (0))k kk k

y uz x (0) (0) (0) 0uz x s.t.

to it is easy to see that the constrained problem (1) is equivalent to the free problem

(1)

(0)0 ( ) : ( (0) (0)) mink

k

f u uz x 0 (0)

( (0) (0)), max ,k

kk

xu uz

(2)

Trivially, u0

is the unique solution of problem (2).

)0(

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Introducing a convergence parameter [0,1[, and recursively solving the optimization problem (2) for increasing t, one gets the following

Theorem B

(2010)

Sequences

and (t ) are monotonically convergent, sequences (z(t)) is convergent and for the respective limits

= lim ,

= lim(t ) and z=lim(z(t)), we have

z

=

z.

Remark 1. For every t, the normed eigenvector of is unique, and its extension to z(t)

is also unique. Therefore, the asymptotic equilibrium attained by immigration control is also uniquely determined.

50 ( )FL t

))t(())0((

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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.

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•

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.

Immigration scenarios and simulations

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In the simulations we considered 3 immigration scenarios

A)

Strongly limited constant yearly immigration (180.000)

B) Constant yearly immigration, corresponding to year 2008 (i.e. 494.289 new immigrants)

C) Immigration controlled by our model, based on the

algorithm moving the age distribution of the population towards a demographic equilibrium (eigenvector of the updated Leslie matrix)


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•

In the model analysis we concentrate on the change of the inverse dependency ratio.

•

In the following figures, for scenarios A), B) and C), we plot against time (years) the

Inverse old-age dependency ratio =

and the

Total population size,

then, for scenario C), the

Yearly total immigration.

Number of people aged 15-64Number of people aged 65 and over


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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

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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.





Scenario (B): constant yearly immigration –

494.289

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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.





Scenario (C): Control model, initial immigration = 450.461

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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.



Figure 7. Total population size,

for scenario C), plotted against time (years)control model, initial

immigration= 450.461, time horizon= 200 years

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Figure 7 shows the long-term stabilization of the total population size for Scenario C.



Figure 8. Yearly total immigration,

for scenario C), showing long-term stabilization of immmigration policy, initial

immigration= 450.461,

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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.



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

ratio.

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References•

M. Angrisani, A. Attias, S. Bianchi, Z. Varga

(2004). Demographic dynamics

for the pay-as-you-go pension system. Pure Mathematics and Applications, 15 (4), 357-374

• Attias A., Equilibrio demografico e flessibilità del sistema pensionistico, Atti del XIII Convegno di Teoria del Rischio (2007), pp. 5-37

•

Leslie, P. H. (1945). On the use of matrices in certain population mathematics. Biometrika, 33, No 3, 183-212

•

Leslie, P. H. (1948). Some further notes on the use of matrices in certain population mathematics. Biometrika, 35, No 3-4, 213-245

•

Varga, Z. (1984) Un modelo

en la dinámica

de poblaciones. Notas

de Matemàticas

de la Universidad Central de Venezuela, 84-ED-03 1-40

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