EPTIn this study, I’m going to combine two economic models to answer a simple question: how could you model whether or not somebody is working themselves to debt? Here, we will look at two models: labor and health.
The labor model of consumer theory is as follows: It’s based on the idea that you are given a finite amount of time every day, and that you can choose how to allocate this time between work and leisure (meaning, not work).
In most cases, we would say that we want to maximize Utility, subject to how much of each good we have. In this case, however, we have a budget constraint of:
pC + wR = M + wL
where p is the price of goods, C is how much we consume of goods, w is our hourly wage, R is our time of leisure, L is our time of work, and M is non-wage based income.
We can simplify all those variables to say that whatever money we earn + money for how much we work equal the amount of money we “lose” for goofing off + the amount of stuff we can buy.
However, economists assume that most people get 8 hours of sleep a day-therefore they have 16 hours to allocate to work and leisure. Therefore, one can substitute in for work as:
L = 16-R
Therefore, a typical labor problem would be:
Maximize U(C, R) s.t. pC + wR = M + w(16-R)
And with two variables, we can use utility maximization to find the answers.
Now, the health model is interesting. It is based off of the idea that people would rather not die-which is a fairly reasonable idea to make, I think. It introduces a new variable into the equation, however, which is π. Π’s meaning is of probability, i.e. the probability that you catch a disease and die.
The health models, while there are many, many of which are complex, generally follow the following formula:
U(x1, x2) = π1V1(x1)(π2V2(x2)) s.t. M = p1x1 + p2x2
In other words, the overall benefit you derive from picking how much you invest into protecting yourself from diseases x1 and x2 is equal to the chance that something you do improves your survivability for each disease. It assumes that you have a certain amount of funding that you must choose to divide among x1 and x2. This funding could take the form of anything, from eating properly to seeing the doctor to chemotherapy to wearing a condom.
Of course, if you wanted to, you could make the function far more specific and binding to a particular subset. But that would require like another 30 derivatives, which would be a pain in the ass. And because I’m trying to avoid math as much as I can, I’m going to do that.
In any case, how do we relate the two equations? The answer is, quite simply, through a form of substitution.
The idea is this: working too hard has an impact on our survivability.
Maximize U(C, R) s.t. pC + wR = M + w(16-R)
Maximize: U(x1, x2) = π1V1(x1)(π2V2(x2)) s.t. M = p1x1 + p2x2
Maximize: U(x1, x2) = π1V1(x1)(π2V2(x2)) s.t. M = p1x1 + p2x2
Therefore, we can assume that our utility should be a function of C and R, i.e. the benefits we get from working and the benefits we get from not working. Because the stuff we can get from working will benefit us, it will be in our final utility function. Because leisure will improve our survivability, we can change the health function by substituting in R for x1, so we get:
U(R, C) = πV(R) + C
Which means: the amount of total utility is based on how much time we can relax (and therefore our health) and how much stuff we can buy. Because we are using R and C, our limiting factor is:
pC + wR = M + w(16-R)
so our final expression is:
U(R, C) = πV(R) + C s.t. pC + wR = M + w(16-R)
What does this mean? It means that there is a specific point for all individuals where they can self-select their own needs to figure out how much they have to work and survive.
Now here’s an interesting application that we can look at: Suppose we were looking at the disease of AIDS. The reason AIDS infection rates aren’t really decreasing is because people don’t see AIDS as an immediate problem-their immediate concern is food and providing for their children and not getting killed by paramilitary forces or getting malaria, etc. So we spend billions and billions on AIDS awareness, but are we really getting any benefit from it?
Let’s see what this implies from our equation: Because we have C (meaning everything but AIDS) versus AIDS, we can see very clearly that the probability of dying from AIDS is not high enough to cause people to put more energy towards preventing infection from AIDS.
Another idea is that the reason AIDS is so prevalent in Africa is due to the overwhelming poverty there, and that by donating money to these families we could intrinsically wipe out AIDS. Well, according to our formula (which is an oversimplification of more accurate models) we are increasing C. Would the person thus want to get more leisure? The answer, of course, is not necessarily.
There is one thing we could do, however, that could help deal with AIDS. Consider the health function:
U(x1, x2) = π1V1(x1)(π2V2(x2)) s.t. M = p1x1 + p2x2
Let’s say one disease is AIDS and one disease is malaria. Malaria kills faster, is easier to get, and has a higher mortality rate (per annum) than AIDS. As a result, we can assume the probability for getting malaria and dying is far higher than that of AIDS. As a result, we can expect that most people would put their energy into not getting malaria, rather than worrying about AIDS. Even from an evolutionary standpoint, this makes sense: people who survive malaria will be able to have more viable offspring, as we see with the case of sickle cell anemia.
Now, suppose we gave a lot of energy into fighting malaria, so we increase x2. By increasing x2, we give the person more incentive to put their energy towards fighting x1. In other words, by increasing malaria protection, we can in turn see a decrease in AIDS infection (by people spending more effort to not get it). Consider how in countries where there is almost nil risk of getting malaria, like in the US, people are very worried about AIDS. That’s because so much energy has already been devoted towards x2 (malaria and other diseases) that we are able to have the luxury to worry about a disease that kills gradually and slowly.
