On Course for a Cleaner Hudson

New York Times article "On Course for a Cleaner Hudson" (September 1st, 2010) discusses the best way to clean up polychlorinated biphenyls (PCBs) in the Hudson river. In a previous article, "Don't scorn Germany and Japan; learn from them", Corrections has extolled the near-incomprehensible properties of exponential growth. However, when the discount rate is greater than the growth rate, this may not be the case.

The E.P.A. now says the dredging might end up taking 10 years, which is fine. A job done slowly and right is better than one altered or abandoned.

Fortunately, we will not have to wait until the job is completed to see good things happen. When PCB concentrations start falling in the river, they decline in fish. This means the benefits of the project will start being felt long before the last load of toxic mud is pulled up from the bottom. This is the strongest rebuttal to G.E.’s old argument that the answer is to let the carcinogens lie in the river, decaying on their own.

If the cleanup of a river that has been tainted for 60 years and counting takes a few years longer than first planned, nobody should be complaining.

We can model the type of situation that the New York Times is suggesting. Specifically, we might have something we desire, say the Hudson River's stock of fish. We have two options for the same cost: quick cleaning and thorough cleaning. The growth rate of the stock of fish is increased to a higher long-term level due to thorough cleaning, but more slowly. It is increased to a lower long-term level due to quick cleaning. The difference between the changing growth rates is depicted graphically below (click to enlarge).

What would this mean for fish stocks? Normalizing our starting stock to one, we can display this graphically as well (click to enlarge)

However, we value current benefits more than we value future benefits--we discount exponentially. If our discount rate is greater than our growth rate, then it's quite possible that the net present value of "quick" is more valuable than "thorough". While exponential growth is a powerful force, exponential discounting may make slower growth now preferable to rapid growth later.

One way to illustrate this is to depict how much we value the first period, the first two periods, the first three periods, out to as many as we please. We do so graphically below (click to enlarge). As one can see, while the gap is closed, the net present value from today including any period in the future will always indicate quick growth as better.

Corrections simply notes, therefore, that a job done quickly is not always worse than a job done thoroughly--there are distinct tradeoffs even when considering exponential growth.