Simplified models are hard
In Engineering, when we sit down to try to describe a simplified version of a problem, there are usually some obvious concepts to start with. Energy, momentum, thermal conductivity, mass density and all sorts of other descriptions we can use that have well developed physical laws. Not so much in social sciences though.
To play with techniques in stochastic optimization of a dynamic system I'm inventing a 4 good economic model: Infrastructure, Human Development, Population, and Business. I'm trying to produce something that captures qualitatively some of the things we know about the way economies behave in aggregate. For example, too little infrastructure and business productivity suffers, too little human development and business productivity suffers, as human development increases, population growth rates generally decline... and so-on.
As I go along trying to write ODEs for these 4 goods I keep thinking of complications that I'm ignoring. For example, population is not a homogenous thing. There's an age distribution, and a distribution of average human development by age. And these things are correlated through time. When we invest money in human development (say through education, healthcare, nutrition and soforth), it affects different people in different ways, perhaps we reduce the death rate among infants, but also among the elderly... When the elderly die, they take with them the knowledge and other forms of human capital that they have, but when babies are born they are totally undeveloped. So in the absence of investment, having babies reduces the growth rate of per capita human development, but increasing lifespans increases the growth rate... until the average lifespan stabilizes. There's not much we can do by monetary investment to make people live 300 years for example. Education will generally affect the younger more than the older generations, and the younger generations are more involved in business. But education can also increase lifespans as people avoid dangerous things. There are different types of business, some may increase production of goods at the expense of requiring dangerous types of work... and therefore having reducing effect on human development.
So it's a hard problem to simplify, and doing so requires us to decide what things we're interested in modeling, and what things we're not that interested in. I don't want to predict the future of a real economy, I want to see how sensitive a toy model of an economy that "feels reasonable" is to things like natural disasters, and what investment strategies can do to mitigate the effects. I don't claim that I'm going to learn something that can be applied directly to real economic systems, but if the model isn't too stupid we should get some qualitative predictions that can be applied indirectly.
For example, if the model is applied to some place like Haiti or Ghana, which strategy would be favored: build secure infrastructure quickly to avoid loss of life and protect the people and help grow business, focusing on human development through education and vaccination later, or build shoddy buildings now, focus on educating people, and use that education to jump start business which can produce the goods and services needed to upgrade the infrastructure later?
if the answer is clear cut one way or another, what assumptions that went into the model cause that behavior? Are those assumptions valid for such undeveloped countries?
The advantage of working with differential equations is that it's much easier to think about what might happen in a short time interval than to think about the long term effects of policies. But I'd like to avoid the complication of modeling the age distribution and the human development distribution, and the wealth distribution as functions of age... I want simplified dynamics of averaged quantities, and simplifying in a reasonable way is hard.