There are a few issues associated with the focus on the ideal case with the main one being that it isn't the problem we actually try to solve. It is much more common that we attempt to solve a simpler sub-problem. Reality rarely gives us all the information we need for the ideal case, so while we might want to believe in the Platonic ideal for the problem, we work with mundane versions.
Earlier I asked what you might want for breakfast 10 years from now. Few people will bother to collect this information based on their belief that it doesn't really matter. There is little doubt you will want breakfast of some kind on that day, but the precise preferences you have don't have to be known in great precision. If you are a vegetarian, we might exclude sausage from the list of your preferences and from that exclude the means for getting sausage to you. If you are vegan, the list of exclusions is even more severe seemingly simplifying the problem. If you always have orange juice for breakfast, that is another kind of simplification. In both of these cases, we can set part of the solution to the overall problem without any further thought. For a more complicated example, consider what we would do if you tend to rotate between typical breakfasts like omelets, pancakes, and oatmeal. That would be enough to set probabilities at a minimum and that might be enough to get close to a solution.
Simplifying the economic problem this way isn't without consequences, though. The primary one is easy to explain. If I don't know what you want for breakfast 10 years from now, but do have enough information to make reasonable guesses, I might guess right or I might guess wrong. If my planned solution to the problem includes enough resources to provide for omelets, pancakes and oatmeal you will get what you want 10 years from now. Unfortunately, though, I've wasted resources in accomplishing that because you'll pick one of them and not the other two. Those are resources I might have put to better use serving preferences for someone else. What happens on the day when I fail to serve their preference and spent too many resources serving yours? How long would it be before someone thought that you had an unfair advantage in the plan? To avoid that, maybe I should just ensure enough is produced to cover all the probabilities so no one complains? That just ensures most of what gets produced ISN'T optimal. Perishables that don't get consumed get wasted and we might think we can live with that, but what about spending the resources used to produce that wastage on coming up with truly new products and services? What about innovation? Would there be any?
Yet this is pretty close to what we do when optimizing our resources for our families. We reduce the big problem to a smaller one and simply try to do our best. I don't know what you want for breakfast 10 years from now. Neither do you probably. Who cares? What do you want for breakfast tomorrow? If you want something exotic that I needed years to prepare to get, too bad. You aren't going to get it because I'm not omniscient. Next problem please.
Unfortunately, this limitation isn't enough to be convincing. It is vulnerable to the 'We are humans' counterargument because our computers aren't human. If we augment our data collection and processing capabilities, surely we can solve the harder version of these problems with more variables in them. Actually, we can't and the worst thing about it is that a solution could be right in front of us and we wouldn't know it in advance. The problem doesn't lie with knowing we have a solution. It lies with knowing we have an OPTIMAL solution. Remember that we are trying to optimize in the ideal case and that means we are trying to avoid waste. That is a big part of what we mean by economic efficiency, after all.
If you know the mathematics, you might be tempted to say it is just a matter of taking differences between neighboring solution attempts and using them to approximate the derivatives in a gradient. If you know a bit more, though, you'll remember that we must be careful about assumptions regarding continuity of functions. The ideal case solution is actually a function of many variables with time as a parameter of that function's curve. There is no reason to assume the function is continuous in all the variables, let alone continuous over long periods of time. In fact, it is probably a terrible assumption. How many pregnant women continue to want pickles and ice cream after giving birth? Our lives are full of discontinuities, so our preferences should be too.
It is worse than that, but I'll save the next material for part 3. We must face two inconvenient truths forced upon us by the mathematics of multi-variable problems.
No comments:
Post a Comment