boom or bust: a model of the economic impact of the baby boomers les fletcher brad poon april 29,...
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Boom or Bust: A Model of the Economic Impact of the Baby Boomers
Les FletcherBrad Poon
April 29, 2003Math 164 Scientific Computing
Overview
Introduction Questions Model Description Results / Analysis Conclusion
Introduction The Baby Boomers resulted in a rapid increas
e of the world’s birth rate, which is only now skewing the elderly / youth ratio eg. United States, Japan
We want to analyze the impact of this disproportion, particularly in the economic sector
Use actual U.S. demographic data to accurately model it’s population trend ie. U.S. Department of Labor 2001 census report (
http://www.bls.gov/cex/csxann01.pdf)
Questions For what initial injection of a baby
boomer birth rate will cause a significant collapse in the economy?
Should we encourage the elderly to do volunteer work in the economy as a means to remediate the problem?
Should we push back the age of retirement?
Initial Model Break up population into sub-
populations based on age Youth (0-9) Adolescent / Young Adult (10-19) Adult (20-64) Elderly (65-death)
Use time-continuous differential equation to model population change
Initial Model (cont.)
Youth (X)
Adolescent (Y)
Adult (Z)
Elderly (E)
XcbbaZdt
dX])1([ YeddXcb
dt
dY])1([])1[(
ZgffYeddt
dZ])1([])1[( hEZgf
dt
dE ])1[(
a : birth rate by adults
b : death rate of youth
c : ascension rate to adolescents
d : death rate of adolescents
e : ascension rate to adults
f : death rate of adults
g : ascension rate to elderlyh : death rate of elderly
Initial Model (Results)
0 50 100 1500
0.5
1
1.5
2
2.5
3
3.5
4x 10
9 Population vs. Time
Time (years)
Pop
ulat
ion
YoungAdolescentsAdultsElderlyTotal
• Population growth is exponential!
Modification to Model
Need to add some sort of “carrying capacity” limit to the model
Solution: Add a separate resource function and make births and deaths dependent on amount of resources available
Similar to consumer resource models
New Model with Resource Dependence Define new resource function:
Therefore, at each time step we will have: Resources consumed: Resources available:
)Ep-(c+)Zp-(c+Y)]+)(Xp-[(cdt
dREEZZXYXY
c : consumption rate of resources
p : production rate of resources
Ec+Zc+Y)]+(X[c EZXYrcR(t)ra
Population Dependence on Resources The population birth and death rates
will now vary as a function of available resources:
XcRbRbZRadt
dX]))(1()([)(
YeRdRdXcRbdt
dY]))(1()([]))(1[(
ZgRfRfYeRddt
dZ]))(1()([]))(1[(
ERhZgRfdt
dE)(]))(1[(
}
Population Dependence on Resources (cont.) We need to define limits for the birth
and death rates with respect to the availability of resources For example, if there are no resources
available, birth rates will go down and death rates will go up
Population Dependence on Resources (cont.) Solution: Use a step-function inverse tangent!
If (rc/ra < 1), then A = some lower limitelse A = some upper limit
11tanmodifierdeath 1
ra
rcsA
11tanmodifierbirth 1
rc
rasA
A : max change
s : sensitivity to change
Results (control group)
0 50 100 150 200 250 3000
1
2
3
4
5
6
7x 10
4 Population vs. Time
Time (years)
Pop
ulat
ion
YoungAdolescentAdultElderlyTotal Population
0 50 100 150 200 250 3000.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2x 10
7 Availability of Resources and Need for Resources vs. Time
Time (years)
Res
ourc
es
NeedAvailibility
Results (baby boom)
0 50 100 150 200 250 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
4 Population vs. Time
Time (years)
Pop
ulat
ion
YoungAdolescentAdultElderlyTotal Population
Now, at time t=100 years, increase the birth rate for 5 years to simulate a “baby boom”
0 50 100 150 200 250 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
4 Population vs. Time
Time (years)
Pop
ulat
ion
YoungAdolescentAdultElderlyTotal Population
6x original birth rate 7x original birth rate
Population dies!
Should we encourage elderly to volunteer?
• Increase production rate pE of elderly at time of baby boom (t = 100 years) to simulate volunteering:
0 50 100 150 200 250 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
4 Population vs. Time
Time (years)
Pop
ulat
ion
YoungAdolescentAdultElderlyTotal Population
0 50 100 150 200 250 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
4 Population vs. Time
Time (years)
Pop
ulat
ion
YoungAdolescentAdultElderlyTotal Population
50% more productive
Population still dies out
75% more productive
Population recovers!
Should we push back age of retirement?
• Decrease ascension rate of adults to elderly so that the population is more productive for a longer time
Extend retirement age to 70
Population still dies out
Extend retirement age to 90!
Population recovers!
0 50 100 150 200 250 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
4 Population vs. Time
Time (years)
Pop
ulat
ion
YoungAdolescentAdultElderlyTotal Population
0 50 100 150 200 250 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
4 Population vs. Time
Time (years)
Pop
ulat
ion
YoungAdolescentAdultElderlyTotal Population
Conclusion We were able to construct a population model
such that we could “tweak” parameters to simulate various economic recovery policies
Limitations to our model: Baby boom did not actually reflect a youth / elderly
disparity ratio, which was our original intention Death rates are dependent on shared resources, as
opposed to resources specific to each age group No immigration / emigration factors are taken into
account, which is a major factor in population trends