In the economic geography stuff, for example, I started with some vague ideas; it wasn’t until I’d managed to write down full models that the ideas came clear. After the math I was able to express most of those ideas in plain English, but it really took the math to get there, and you still can’t quite get it all without the equations.What Krugman says serves as a useful piece of advice. In shorter terms, the mechanism of mathematics provides the insight to our understanding. Unless you work the math, the insight may never come, simply because it captures and retains the bookkeeping and juggling of ideas for our over-taxed brains. Often times you would never know if certain effects canceled out, compounded, or came out negligible unless you took the time to formalize the arguments.
Take a look at one of Krugman's books on economics and geography. One area of interest he had concerns the size and spatial distribution of city and town populations, and specifically the organization of edge cities. Many had observed a general behavior of very few large populations and progressively more smaller cities. This has the moniker of Zipf's Law -- a heuristic that seems to match the data from a rank histogram of the USA:
P(Size) = k/Sizen, where n=1
Granted some people might consider this completely intuitive from the start and require no insight beyond this point to make more elaborate arguments. Or you can take this empirical result and try to understand why it occurs by diving further into the math (Mitzenmacher and Gabaix). In one of his textbooks ("The Self-Organizing Economy"), Krugman had invoked a preferential attachment argument due to Herbert Simon to explain the historical city population growth.
My own understanding derives from a straightforward application of dispersion to growth rates of the measure of interest. For the case of oil reservoirs, a maximum entropy dispersion of material drift velocity during its formation can generate such a dispersion. The solution to this leads to a probability distribution function
P(Size) = 1/(1+c/Size)
This looks something like the Zipf-Mandelbrot variation which includes a constant limiting term to prevent a singularity at the origin (in ecology this is also known as a relative abundance distribution).
We can illustrate the similarity between the distribution of city population sizes (from USA census data) with the distribution of discovered oil reservoir sizes in the USA (from the Baker and Gehman data). See to the right for a ranked histogram of the two sets of measures. I fit a dispersive aggregation model for each set of data with the single parameter c describing the characteristic aggregation size. Most interesting in the aligned data set is the multi-level correspondence between the number of cities and the number of reservoirs at varying densities. First note that the number of very large cities and very large reservoirs is comparable, about 9 cities (not urban areas) over a million people and about 13 or 14 oil reservoirs over a billion barrels in recoverable reserves. From there, the ratio holds relatively steady, for every city of a specific size numbered in thousands, you find an oil reservoir with that same size in million barrels that you can "attach" to that city. So you find that around 200 cities have 100K or more in population and about 200 reservoirs have 100 MB or more in oil. This works for awhile until the number of small cities starts to overtake the number of small reservoirs as counted by Baker.Applying a sanity check to the 1000x ratio, means that each "reigning" citizen of a large city will use about 1000 barrels (100 MB/100K) of natively-supplied oil over the course of time. I use the term "reigning" to indicate that each city's population has followed a slowly growing equilibrium that will asymptotically reach some carrying capacity with births/immigration balanced by deaths/emmigration, whereas oil has a finite limit not at all related to a carrying capacity. So reigning essentially means a steady-state citizen, and that equivalent person has used 1000 barrels or 42,000 gallons of American-sourced unrefined oil over the past 150 years.
I find this exercise useful, if for nothing else, that it can give people a feel for how many reservoirs that we have remaining. Think of how many sizable cities we have, and that amounts to how many equivalently-sized reservoirs that we have to essentially feed those cities. Eventually, all these reservoirs will turn into ghost towns as they deplete and become shut-in while the cities will remain. Most people do not have a feel for what 300 million people means yet they can start to comprehend thousands of reservoirs and cities, and the significance of those numbers. It also helps to put the "Drill, Baby,Drill" nonsensical mantra into context: think in terms of how many reservoirs we would need to discover to replace the ghost reservoirs that will crop up -- essentially one for every city if we want to depend on a captive native resource.
As influential as Krugman remains among rational economists, I wonder when he will really start hitting the real problems of constrained resources that this country (and the world) faces. Economists have the habit of theoretically deferring to the substitution of one resource for another as soon as we reach a constraint. However, without clear technological alternatives, this constraint looks ominous and would make Krugman look even more like a contrarian than he normally appears to the supply-siders. I know Krugman could comprehend the math, but I don't know whether he wants to go there.