Too many engineering papers I review get this basic economic concept totally wrong.

Wolfgang Pauli once remarked about a paper “that’s not right; it’s not even wrong”. His point was that in order for a paper to be wrong, the authors need to come to incorrect conclusion based on a correct premise. But if the authors start from an incorrect premise… well… then the paper isn’t even wrong.

Recently, I was once again asked to review a paper in which the authors claim to have achieved an optimal economic objective by minimizing the average LMP in a hypothetical ISO when demand varies as a function of price. On the surface this might seem like a reasonable goal–I think it goes without saying that reducing costs is what consumers seek–if the average LMP is minimized then wouldn’t one expect the cost of electricity to consumers is also minimized?

Unfortunately, this is not always true, and it seems this is a mistake engineers make more often than one might expect. In truth, cost is only minimized by minimizing prices when demand is not too sensitive to price.

Here’s why: Suppose demand increases fractionally in proportion to the fractional reduction in price, e.g., the demand is doubled when the price is halved. Economists define this as the demand elasticity

\[\eta = \frac{d \log q}{d \log p} = \frac{p}{q} \frac{dq}{dp}.\]

We expect $\eta$ to always be negative because demand rises when prices fall and vice versa. When $-1<\eta<0$, demand is said to be inelastic because when the price increases, the demand decreases trivially. However, when $\eta<-1$, demand is said to be elastic because when the price increases, the demand decreases significantly.

Now consider how the cost $c=pq$ changes with respect to price, i.e.,

\[\frac{dc}{dp} = p \frac{dq}{dp} + \frac{dp}{dp} q = \eta q + q = (1+\eta) q\]

When $-1<\eta<0$, the cost increases with increasing price, just as we might expect. But notice that when $\eta<-1$, the cost decreases with increasing price, which is the opposite of what one might expect.

The consequence of this is that when inelastic demands are considered, lowering electricity prices does indeed result in decreased costs to consumers. As a result, an optimization which uses a price-minimizing objective function may indeed result in reduced consumer costs, provided demand always remains inelastic.

However, when considering the effect of introducing demand response technology in electric power system loads, we may be encounter conditions in which demand is more elastic, and possibly to the point where $\eta<-1$. If a new technology significantly increases the price elasticity of loads such that demand goes from being inelastic to being elastic, then an objective function that minimizes prices will not result in a minimal costs. In fact, quite the opposite, the cost may be maximized.

In general, it is not correct when an objective function seeks to minimize prices to achieve an economically optimal outcome. It is more correct to consider costs, and if demand is sufficiently elastic, it is even more correct to consider surpluses, i.e., the difference between cost and benefits, despite the difficulty of measuring the latter. But that is a subject for another time.