午夜剧场

In economic theory, we analyze phenomena using models built from only the essential elements of real-world occurrences. Taking the prices of products in a supermarket as an example, we first model the behavior of consumers who buy products by deciding what and how much to buy to maximize their satisfaction (utility) within the limits of the money in their wallets (budget). We then model the behavior of producers who sell products by deciding what and how much to produce to maximize their profit. The price at a "competitive equilibrium," where the amount consumers want to buy (quantity demanded) and the amount producers want to sell (quantity supplied) match in the model, corresponds to the price you see at the supermarket. For real-world analysis, it is problematic if a competitive equilibrium does not exist in the model, so we need to know under what conditions a competitive equilibrium exists.

Recently, I have been researching the existence of competitive equilibrium in the presence of "bads"—things like garbage or contaminated items that decrease satisfaction (utility) when possessed. To show that a condition identified as essential for the existence of competitive equilibrium is truly indispensable, one must create a concrete example where competitive equilibrium does not exist because that specific condition is not met. When I was a graduate student, I created examples solely through manual calculations, but now I enlist the power of the internet. When I input a function and search on Google, a graph appears, making it easier to create functions with the desired properties. Furthermore, while there were limits to manual integration, thanks to sites that return primitive functions, I less frequently have to abandon the creation of an example halfway through because the calculations are too complex.

One hundred years ago, the mathematicians Ramanujan and Hardy are said to have asked MacMahon, a veteran with extraordinary calculating power, to perform calculations to verify the accuracy of one of their approximation methods. At that time, only top-tier mathematicians could utilize top-tier calculating power, but today, with the development of the internet, researchers like myself can use extraordinary calculating power for free.

The other day, I calculated a primitive function on a certain site and created an example where a competitive equilibrium does not exist when bads are present. A fellow researcher told me, "I don't intuitively understand why it doesn't exist," which made me anxious that I might have made a mistake in the integration by substitution I calculated myself. However, after recalculating many times, it seems to be correct. I am quietly pleased with myself for having created a good example for the first time in a while.