boom or bust: a model of the economic impact of the baby boomers les fletcher brad poon april 29,...

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