Tomoki Inoue: A Calculation Sidekick

2019/03/08

In economic theory, we analyze using models (representations) created with only the essential elements of real-world phenomena. 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 satisfaction (utility) within the limits of the money (budget) in their wallets. We then model the behavior of producers who sell products by deciding what and how much to produce to maximize profit. The price at "competitive equilibrium," where the amount consumers want to buy (quantity demanded) and the amount producers want to sell (quantity supplied) in the model match, 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 it is necessary to know under what conditions a competitive equilibrium exists.

Recently, I have been researching the existence of competitive equilibrium when there are "bads"—things like trash or contaminated items that decrease satisfaction (utility) when possessed. To show that the conditions identified as essential for the existence of competitive equilibrium are truly indispensable, one must create a specific example where competitive equilibrium does not exist because only those conditions are not met. When I was a graduate student, I created examples by doing manual calculations relentlessly, but now I rely on the power of the internet. By entering a function and searching on Google, a graph appears, making it easier to create functions with the desired properties. Also, while there were limits to manual integration, thanks to sites that return primitive functions, I less often give up on creating examples 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 calculation skills, to perform calculations to verify the accuracy of one of their approximation methods. At that time, only top-tier mathematicians could utilize top-tier calculation skills, but in today's world of the developed internet, researchers like myself can use extraordinary calculation power for free.

The other day, I calculated a primitive function on a certain site and created an example where competitive equilibrium does not exist when there are bads. A fellow researcher told me, "I don't intuitively understand why it doesn't exist," and I became worried that I might have made a mistake around 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 after a long time.