This is a project I worked on and wrote when I was a first year PhD student – roughly around 2016-2017. It was intended as a blog form piece. I’ve left the original text as is – there may be typos or occasional incomplete logic, but surely this will be useful to someone.
Marriage and divorce represent one of the most dominant forces in national social interactions and of the modern day. In US popular culture the divorce rate has been the subject of media discussions and generally societal for a decade, and with it comes supposed justification for a myriad of social change arguments and debates.
For this piece, I briefly outline the makings of a simple system dynamics model that aims to reproduce the marriage and divorce rate through simple stocks and flows, and with particular attention paid to basic population dynamics in the period from the year 2000 to the year 2014. I propose that much of the explanation for variance in the divorce rate, commonly calculated as the number of divorces in any given year divided by the number of marriages for any given year can be attributed to small fluctuations in the rate of marriage and the rate of divorce for the population and that they are independent processes.
Becker’s (1973) seminal paper on the theory of marriage and divorce applied economic principles for the first time to marriage and divorce dynamics. In his papers—a multi-part collection—he applied a number of economic theories such as the theory of preferences, planned behavior, and others to explain the market forces of marriage and divorce. In his mathematical and economic model, he endows various types of market actors, such as males or females with attributes, preferences, and production functions that include labor and non-labor products and assumes that the marriage and divorce market will maximize utility for all participants, not any one individual.
Others within the same period model the marriage market so as to be the result of social and psychological forces. Levinger (1976) takes that approach and includes the concept of cohesiveness as the central mechanic by which both marriage and divorce take place.
Research in this area stagnated in the following decades, though national marriage and divorce statistics continued to be published by agencies such as the US Census Bureau and eventually the Center for Disease Control. This led many to include other data-based theories to marriage and divorce including social norms (Ishida, 2003), cohort phenomenon (Elwood, 1979), employment levels of both men and women, purchasing power and other factors.
More recently, an effort was made to understand this important process through modeling and simulation, much of which is agent-based (Matthews et. al., 2013) (Brien et. al, 2006).
We used Census Bureau and Center for Disease Control data to gather total population numbers for the United States between the year 2000 and 2014. This period was chosen because it would have likely had the best available data during the last few decades of data collection. The data was organized by gender as it was my intention originally to build a model around gender specific population dynamics but it quickly became apparent that this pathway did not necessarily bring any additional value to the modeling effort. For a high fidelity model, better use can be made of the traits of the modeled population, including gender, race, age-group, economic class, and others—all of which would likely have some impact on the localized divorce and marriage rates and especially in the case of the rate of remarriage.
The data collected was incomplete in some areas, and in order to bridge the gap between the necessary data for the model and the available data, some data points were approximated using a linear approximation based on other data points in the time-series for the period in question. This likely introduced some error in the simulation but we assume that it is negligible for now. Table 1 shows population data collected from the Census Bureau for the United States between 2000 and 2014. I also collected the total number of married and divorced persons in the year 2000 to form the default parameters of the model. Table 2 summarizes the data collected from the Census Bureau and the Center for Disease Control.
The model itself is a simple system dynamics model with 3 stocks and 2 flows and a small number of parameters. Stocks are represented by the actual counts of the population, the married population in 2000 and the divorced population. Rates of growth were added using a regression parameter drawn from the yearly number of divorces and marriages for each subsequent year. Additionally, a second rate of marriage and divorce rate increase or decrease was added to estimate the yearly rate of marriage and divorce as it changes over the years modeled. Adjustments were made for missing rate data points by assuming that the rate of divorce and marriage for missing data points reflected the average of the whole population. Figure 1 shows the model as it was represented in AnyLogic—a software package for dynamic system modeling.
The model reproduced the declining marriage/divorce ration—also known in popular culture as the divorce rate and commonly referenced as 40%-50% by the majority of people that discuss it. Figure 2 shows a simple graph of the nonlinear decline in the divorce ratio as an illustration. And, though the difference is that this method calculated the ratio of all married and divorced members of the population (not simply the number of people married or divorced per any given year) we can plainly see that the population ratio of divorced to married individuals in the overall population is actually around 20%-a much more reasonable number.
It was also clear that the reference to a declining divorce rate commonly referenced is an artifact of the changes in simple population dynamics and differing 2nd order rates of change for both the marriage and divorce rates respectively. In our specific example, the marriage rate is increasing and the divorce rate is increasing but the divorce rate is increasing at a slower rate than the marriage rate, hence a nonlinear relationship that starts out as positive and gradually declines over the period of time we reference. Thus, one may argue that the reason for the apparent decline in the popular culture divorce ratio emanates from a second order effect or differences in the various rate of increase or decline of the rates of divorce and marriage rates. Figure 2 illustrates this phenomenon.
This model was as simple as can be made and although it may seem to be generally uninteresting, by modeling the stocks and flows we were able to discover the internal dynamics of a process that was taken for granted. This is a reasonable result made possible by the powerful tool that is system dynamics modeling and simulation, but I do have to admit – modeling processes as differential equations is just not as much as fun as I thought it would be.
Perhaps, the fact that the pattern of divorce and marriage is highly dependent on systems of differential equations which ultimately results in a nonlinear declining relationship could explain why the divorce rate is such a salient topic of popular culture. Oftentimes the topics that are least understood are the ones where nonlinear dynamics and complexities are in effect.
In future model iterations, a number of improvements can be made to the model, including attributes such as age-group, economic status, and separate gender flows.
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