Thursday, August 9, 2012

Beans, carbon credits, and partial derivatives

Anybody remember this little jingle from childhood? (Warning: may be NSFW)
I'm often tempted to use this in Calculus III to help explain the difference between partial derivatives and total derivatives.

If you have a system in which all sorts of quantities x, y, and z and so on are varying, the partial derivative of z relative to y (say) measures the rate that z would change if y was varied and everything else was somehow held constant.  ("The more you fart, the better you feel" - assuming you ate a constant quantity of beans to start with).

The total derivative though (of z relative to x, say) measures the rate that z would change if x was varied and everything else in the system responded in the "natural" way ("So eat baked beans with every meal" - we are not holding the fart rate constant here while changing bean consumption - I don't even want to think about the experimental protocol to achieve that.)

These ideas are not the same and confusing them can lead to some paradoxical policy advice (as in the jingle!).  I was reminded of this when reading an important article about the unintended consequences of emissions trading.: "since 2005 the 19 plants receiving the waste gas payments have profited handsomely from an unlikely business: churning out more harmful coolant gas so they can be paid to destroy its waste byproduct. The high output keeps the prices of the coolant gas irresistibly low, discouraging air-conditioning companies from switching to less-damaging alternative gases. That means, critics say, that United Nations subsidies intended to improve the environment are instead creating their own damage."

Typically, economic incentives are constructed based on partial derivatives - if I increase the price of good g, and everything else stays constant, then the consumption of g will drop.  But in reality, everything else doesn't stay constant.  And the response of the economic system as a whole may be very different - even in the opposite direction - from what simple thinking about one effect in isolation might suggest.  (If g is a Giffen good, consumption of g may rise as its price increases.)

The complexity of the economy's response to incentives at least raises a question about the assumption that "experts" will be able to construct an incentive scheme that stably guides the economy to a "safe" level of greenhouse gas emissions.

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