Boom or Bust: an economic model of the baby boom Math 164 - Scientific Computing Brad Poon and Les Fletcher

May 2, 2003

Abstract We analyze the impact of the baby boomer generation on the economy as disparity between the youth / elderly ratio increases. Our population model incorporates a resource function that modifies birth and death rates to deal with the lack of (or availability of) resources. While our nonconventional consumer resource model does not accurately mirror any specific population, its parameters are easily configurable and flexible. We conclude that, with further modifications and research, this model could be used successfully to help aid economical fiscal policy decision-making.

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Introduction After World War II, the birth rate in the United States soared after weary soldiers returned to their wives and girlfriends. The result became a dramatic disparity between today’s youth and elderly ratio as these ”baby boomers” grew older. This phenomenon also occurred in Japan, which became the first country in history to have an average age of over 40. As a result of this disparity, new economic and health issues have arisen, as the elderly become increasingly more dependent on an increasingly smaller capable workforce. In our model, we hoped to analyze the economic impact of the baby boomers. We used U.S. economic census data to determine our parameters in the hopes that it would accurately mirror the current population situation. We also wanted a model that could be ”tweaked” easily, so we could simulate changes in the population, such as a sudden increase in the birth rate due to baby boomers. We developed our model with the following questions in mind: 1. For what initial injection of a baby boomer population will cause a significant collapse in the economy? 2. Given the above collapse in the economy, should we encourage the elderly to volunteer as a means to remediate the situation? 3. Should we push back the age of retirement and force elderly to work longer?

Initial Model In our initial model, we divided the population into four age groups: young (0-9), adolescents (10-19), adults (20-64), and the elderly (65+). We made a simplifying assumption that only the adults have babies. This meant that the birth rate into the young population is dependent only on the number of adults. We also assumed that there is no immigration into or emigration out of the population. Each age group has a specific death rate at which people in that age group die. The first three populations also accede to the other populations at a certain rate each year. For example, our young age group has ten years of age, so every year a tenth of the youth who do not die become adolescents. The elderly do not have an ascension rate, only a death rate. Given the previous conditions, our equations became:

Youth Adolescents Adults Elderly

dY dt dT dt dA dt dE dt

= aA − (b + (1 − b)c)Y = ((1 − b)c)Y − (d + (1 − d)e)T = ((1 − d)e)T − (f + (1 − f )g)A = ((1 − f )g)A − hE

a is the birth rate, b, d, f, h are the death rates, and c, e, g are the accension rates

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Setting initial conditions and parameters, we obtained the following graph:

Figure 1: Initial Model As we expected, this model gives an exponential growth for the population. We decided this was a good basis for a more complex model.

Resources Model Since an exponentially growing population does not accurately model a realistic population per se, we needed to added some sort of ”carrying capacity” to the model. We decided to add a new resource function R(t) such that the population requires these resources in order to survive. This is similar to a consumer resource model, although our method is somewhat non-conventional in the mathematical setup. At each time step, every person in the population consumes and produces a set amount of resources. Resources consumed at time t Resources available at time t

rc = cY T (Y + T ) + cA A + cE E ra = R(t)

Because the population is dependent on these resources for survival, the birth and death rates are affected by the amount of resources available to the population. In other words, the birth and death rates now become functions of the resource function R(t). When more resources are available than are needed, the birth rate goes up and the death rate goes down, and when there is a lack of resources the opposite happens. Our new model became: 3

dY dt dT dt dA dt dE dt dR dt

ra rc rc ∗ aA − ( ∗ b + (1 − ∗ b)c)Y rc ra ra rc rc rc = ((1 − ∗ b)c)Y − ( ∗ d + (1 − ∗ d)e)T ra ra ra rc rc rc = ((1 − ∗ d)e)T − ( ∗ f + (1 − ∗ f )g)A ra ra ra rc rc = ((1 − ∗ f )g)A − ∗ hE ra ra =

= (pY T − cY T )(Y + T ) + (pA − cA )A + (pE − cE )E

A Revised Model with Dependence on Resources Once we began testing, we realized that this model had a few short comings. When there is an extreme excess of resources, the birth rate sky rockets and the death rates go to 0. Realistically, this does not make sense since people will still die of natural causes, regardless of the resource level. Similarly, the other extreme has the same problem. If there are no resources available in the system, then the death rate shoots off into infinity. If there are negative resources in the system, then the death rates become birth rates! We needed to have a function in which if the need for resources equaled the availability of resources, then birth and death rates are nominal. If the need does not equal the availability, then there must be some ”limit” to the change in the birth and death rates. To this end we decided to adjust the birth and death modifiers with an inverse tangent step-function. This allows us to set the maximum and minimum birth and death rates by adjusting the amplitude of the function. Now our birth and death modifiers look like: ra birthmod = maxbirth ∗ tan−1 (sbirth ( − 1)) + 1 rc rc −1 deathmod = maxdeath ∗ tan (sdeath ( − 1)) + 1 ra sbirth and sdeath are the factors that change how sensitive ra the birth and death rates are to changes in ra rc or rc We chose values for maxbirth and maxdeath so that when (rc/ra) < 1, the most the death rate could decrease to was 25% of the original value, but it could increase tenfold. When when (ra/rc) < 1, the most the birth rate could decrease to was 25% of the original value, but it could only increase twofold.

Results Our new model seemed to perform similar to our initial model. We noticed that the population was gradually increasing as time went on. One thing of interest was that our population’s need was consistently above the availability of resources. We thought this might have been a problem in our model, but then we realized that this was simply modifying our birth and death rates. The fact that the need was greater than the availability meant that our initial death rates were actually too low, and it was correcting for it! This self-correction of our model can be contributed to the see-saw nature of the inverse tangent functions which increase or decrease the birth and death rates appropriately. 4

Figure 2: We see an exponential growth. To answer our primary question, at year 100 we increased the birth rate of the population for five years to simulate a baby boom. We found that a sixfold increase in the birth rate caused a sharp spike in the total population, but then the population recovers (Figure 4). However, a sevenfold increase in the birth rate had some interesting results: the entire population died off (Figure 5)! We concluded that a population in which everyone dies constituted a ”significant collapse in the economy.”

Should we encourage elderly to volunteer? While maintaining the sevenfold increase in the birth rate between years 100 and 105, we increased the production rate of the elderly to simulate volunteering. We find that at 50% increase in production, the population died off(Figure 6), but at 75% increase in production, the population finally recovers (Figure 7).

Should we push back the retirement age? In this case, we simply decreased the rate of ascension from the adult age group into the elderly age group. For example, if we increased the retirement age from 65 to 70 (Figure 8), the new ascension rate would become 1/(70-20) = 0.2. Only at an extended retirement age of 90 did the population recover (Figure 9). Thus, we concluded that increasing the age of retirement would not be a very plausible economic policy.

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Figure 3: Need is always greater than availibility.

Conclusions With our model, we were able to successfully simulate a baby boom generation and find a value for which the population would be unable to support itself. Through further tweaking of parameters, we were able to find new rates at which the population would recover - one, by encouraging the elderly to volunteer and increasing their productivity rate by 75%; two, by increasing the age of retirement to 90 years (Figure 9). The previous condition is a plausible solution, the latter is not. While our results by no means reflect any actual population, we believe that with more accurate data and further research, this population model can aid in economic fiscal policy decision making.

Further Improvements to Model There were several limitations to our population model. First, our model did not actually represent the disparity in the ratio between the youth and elderly as we had originally hoped. Instead, we notice that most of the elderly die out as the baby boomers are born. Part of the problem is due to the fact that the resources are shared among the entire population, which is not a realistic assumption. We should divide the resources by age group, so that when there are a significant number of youth, the elderly do not necessarily suffer as much since the adults are required to take care of their children. Another problem we noticed is that when the baby boomers initially enter the population, the subsequent populations are immediately affected. This is due to our ascension rates, in which a

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certain percentage of the population accedes into the next at every time step. This isn’t accurate when there is a sudden injection of youth, because it doesn’t take into account the several years it takes for these new youths to grow up and become adolescents. This problem can be remedied by creating a differential equation for every age in a population, but this was too tedious and cumbersome for our model, so we divided the population into age groups. Finally, our model does not take immigration and emigration into account. It turns out that immigration is a major contribution to population changes. A better model can be implemented to take these (and other natural) factors into account.

Acknowledgements We would like to thank Professor de Pillis for giving us ideas for our consumer resource model, and offering tips on our model when we were stuck.

References Bloom, David E., and A. K. Nandakumar and Majiri Bhawalkar. "The Demography of Aging in Japan and the United States." March 2001. . "Comparison of the Age Distribution of the US Resident Population in the 200 and 1990 Census." Administration on Aging. Macunovich, Diane J. "The Baby Boomers." Department of Economics. Bernard College, Columbia University. October 2000. . "U.S. Death Statistics."

The Disaster Center.

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U.S. Department of Labor. "Consumer Expenditures in 2001." Bureau of Labor Statistics. Report 966. April 2003. .

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Appendix A: Figures

Figure 4: 6x increase in birth rate. Population recovers.

Figure 5: 7x increase in birth rate. Population dies out.

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Figure 6: 7x increase in birth rate and 50% increase elderly production. Population dies out.

Figure 7: 7x increase in birth rate and 75% increase elderly production. Population recovers.

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Figure 8: 7x increase in birth rate and retirement age of 70. Population dies out.

Figure 9: 7x increase in birth rate and retirement age of 90. Population recovers.

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AppendixB: Matlab Code Population Model using ODE23 in Matlab.

% % % % % % % % % %

populations.m Brad Poon & Les Fletcher Math 164 Spring 2003 Professor de Pillis Last Modified: April 30, 2002 DESCRIPTION: A model of population based on 4 age groups: adult(20-64), and elderly (65-death).

young(0-9), adolescent(10-19),

function yprime = populations(t,y) % Birth and Death Rates a = 0.1144; b = 0.005134; c = .1;

% Birth rate of ’Young’ age group % Rate at which population leaves ’Young’ age group (through death) % Rate at which population leaves ’Young’ age group (through acension)

d = 0.001026; e = .1;

% Rate at which population leaves ’Adolescent’ age group (through death) % Rate at which population leaves ’Adolescent’ age group (through acension)

f = 0.03938; g = .022;

% Rate at which population leaves ’Adults’ age group (through death) % Rate at which population leaves ’Adults’ age group (through acension)

h = 0.32649;

% Rate at which population leaves ’Elderly’ age group (through death)

baby_boom_rate = 1;

% Multiplier for the birth rate to simulate a baby boom

if (t > 100 & t < 105) a = a*baby_boom_rate; end

% Production and Consumption Rates (per person) ry1 = 77; ry2 = 103.6;

% Resource production rate of ’Young’ and ’Adolescent" age group % Resource consumption rate of ’Young’ and ’Adolescent" age group

rz1 = 820.1; rz2 = 774.3;

% Resource production rate of ’Adults" age group % Resource consumption rate of ’Adults" age group

re1 = 102.1; re2 = 122.1;

% Resource production rate of ’Elderly" age group % Resource consumption rate of ’Elderly" age group 11

% Critical resource values, amount of resources needed, amount resources available rc = ry2*(y(1)+y(2)) + rz2*y(3) + re2*y(4); ra = y(5);

% Resources consumed % Resources available

% Modified birth / death rates due to lack of (or abundance of) resources available if (rc/ra < 1) sensitivity_death = 1; max_change_death = .3183; else sensitivity_death = .1; max_change_death = 10; end if (ra/rc < 1) max_change_birth = 1.2732; % -1 / atan(-1) else max_change_birth = 1/(2*pi); end birth_modifier = max_change_birth*atan((ra/rc)-1)+1; death_modifier = max_change_death*atan(sensitivity_death*((rc/ra)-1))+1; % Modify the birth and death rates a b d f h

= = = = =

a*(birth_modifier); b*(death_modifier); d*(death_modifier); f*(death_modifier); h*(death_modifier);

% Population model with resource consumption yprime = [a*y(3) - (b + (1-b)*c)*y(1); (1-b)*c*y(1)-(d+(1-d)*e)*y(2); (1-d)*e*y(2)-(f+(1-f)*g)*y(3); (1-f)*g*y(3)-h*y(4); ((ry1-ry2)*(y(2)+y(1)))+(rz1-rz2)*y(3)+(re1-re2)*y(4)];

The execution and plotting script.

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% % % % % % % % % %

runme.m Brad Poon & Les Fletcher Math 164 Spring 2003 Professor de Pillis Last Modified: April 30, 2002 DESCRIPTION: Script file to run population model and plot results. to run the simulation and see results.

Just type ’runme’ in MATLAB

clear all; close all; % Initial population data based on data 2000 census data from the % Administration on Aging (http://www.aoa.gov/aoa/STATS/Census2000/2000-1990-Pop.html) iyoung iadole iadult ielder iresource

= = = = =

1412; 1448; 5897; 1243; 5014113.4;

years = [1:300];

% % % % %

Initial Initial Initial Initial Initial

young population adolescent population adult population elderly population resource level

% Number of years to run simulation on

% Solve the ODE using ode23 [t,y] = ode23(@populations, years, [iyoung,iadole,iadult,ielder,iresource]);

% Get data for plotting and analysis young_pop adole_pop adult_pop elder_pop resources total_pop

= = = = = =

y(:,1); y(:,2); y(:,3); y(:,4); y(:,5); young_pop + adole_pop + adult_pop + elder_pop;

% Plot of Individual Population vs. Time plot(t,young_pop,’r’,t,adole_pop,’b’,t,adult_pop,’g’,t,elder_pop,’k’,t,total_pop,’k:’); title(’Population vs. Time’); xlabel(’Time (years)’); 13

ylabel(’Population’); legend(’Young’,’Adolescent’,’Adult’, ’Elderly’, ’Total Population’,2)

% Plot of Total Population vs. Time figure; timestep2 = young_pop + adole_pop; timestep3 = timestep2 + adult_pop; plot(t,young_pop,’r’,t,timestep2,’b’,t,timestep3,’g’,t,total_pop,’k’); title(’Total Population vs. Time’); xlabel(’Time (years)’); ylabel(’Population’); legend(’Young’,’Young + Adolescent’,’Young + Adolescent + Adult’, ’Total’, 2)

% Plot of Total Resources vs. Time iresource_level(1:length(years)) = iresource; % Initial resource level figure; plot(t, resources,’r’, t, iresource_level, ’b’); title(’Resources vs. Time’); xlabel(’Time (years)’); ylabel(’Resources’); legend(’Resources’,2)

% Production and Consumption Rates ry1 = 77; ry2 = 103.6;

% Resource production rate of ’Young’ and ’Adolescent" age group % Resource consumption rate of ’Young’ and ’Adolescent" age group

rz1 = 820.1; rz2 = 774.3;

% Resource production rate of ’Adults" age group % Resource consumption rate of ’Adults" age group

re1 = 102.1; re2 = 122.1;

% Resource production rate of ’Elderly" age group % Resource consumption rate of ’Elderly" age group

% Critical resource values, amount of resources needed, amount resources available need = ry2.*(y(:,1)+y(:,2)) + rz2.*y(:,3) + re2.*y(:,4); availibility = y(:,5);

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% Plot of availability of resources and need for resources vs. time figure; plot(t, need,’r’, t, availibility, ’b’); title(’Availability of Resources and Need for Resources vs. Time’); xlabel(’Time (years)’); ylabel(’Resources’); legend(’Need’, ’Availibility’ ,2)

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