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In my introductory microeconomics class, a problem set had the following question:

If the price elasticity of demand for Snickers bars is 1, what will a 1% increase in its price do to the total amount spent on it (not quantity purchased, but total amount spent) each week?

The answer in the problem set solution released later was as follows:

It will leave it unchanged because the percentage change in quantity demanded will exactly offset the percentage change in price.

My problem with this is that isn't total revenue (and therefore total expenditure) supposed to be maximum when the demand is unitary elastic? If so, shouldn't changing the price when demand is already unitary, decrease revenue/expenditure?

I've also tried numerical examples. If the price of a bar is \$2 and 100 units are being purchased, the expenditure is \$200. If the price increases to \$2.02, the quantity should fall to 99. Now the expenditure would be \$2.02 * 99 = \$199.98, which is lower than \$200.

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This is an excellent question (and congratulations for spotting an apparent inconsistency in your problem set!) To answer it, we need to distinguish between two slightly different concepts: the arc elasticity and the point elasticity.

The arc elasticity of demand can defined as the percentage change in quantity demanded divided by the percentage change in price. (Other definitions are available.) For instance (to borrow your example), if initially $P = 2$ and $Q = 100$, and later $P = 2.02$ and $Q = 99$, then the arc elasticity of demand is:

$$\frac{\%\Delta Q}{\%\Delta P} = \frac{\frac{100-99}{100}}{\frac{2-2.02}{2}} = -1$$

Notice that to calculate the arc elasticity of demand we need two price/quantity pairs.

In contrast, the point elasticity is defined slightly differently: it is the limit of the arc price elasticity as the size of the change in price goes to zero. In other words, it is essentially the same as the arc elasticity, but for a infinitesimally small price change. As a result, the 'old' price/quantity and 'new' price/quantity are essentially the same, so to compute the point elasticity we need only one price/quantity pair.

A bit more formally, consider a small price change $dP$ that leads to a small change in demand $dQ$. The point elasticity is given by

$$\frac{\%\Delta Q}{\%\Delta P} = \frac{\frac{dQ}{Q}}{\frac{dP}{P}} = \frac{dQ}{dP}\frac{P}{Q} $$

These preliminaries out of the way, we are now in position to answer your question.

  • Under standard assumptions (downward sloping and differentiable demand), revenue is maximised when the point elasticity equals 1. (You may wish to verify this!)
  • However, at the revenue maximising point, the arc elasticity need not be one. Conversely, if the arc elasticity is one, revenue need not be maximised (as your example indicates).
  • Insofar as the point elasticity is a good approximation for the arc elasticity (i.e. when considering small prices changes), the arc elasticity should be close to 1 at the revenue maximising price/quantity.
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