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Why is it that when demand is inelastic, price and total revenue move in the same direction, but when demand is elastic, they move in opposite directions? I understand in terms of consumer response, but why does it happen mathematically?

Edit: Algebra based eco class using midpoint method aka arc elasticity.

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  • $\begingroup$ The elasticity formula is measuring the degree of responsiveness of quantity demanded to a change in its price. Key Assumption Here: Law of Demand. Case 1: If that change in price leads to a bigger change in Quantity demanded, then lowering price leads to a much greater increase in Quantity demanded, so your total revenue goes up. Case 2: If that change in price leads to a lower change in Quantity demanded, then increasing price leads to a lesser decrease in Quantity demanded, so your total revenue goes up. $\endgroup$
    – Rumi
    Oct 29, 2021 at 2:15
  • $\begingroup$ You may want to think of it in terms of slopes. Think Elastic is flatter slope and inelastic as steeper slope. Draw any two points on the demand curve and look at how because of the slope the corresponding changes look like. The math then follows. $\endgroup$
    – Rumi
    Oct 29, 2021 at 2:17

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Arc elasticity using midpoint method is: $$ \frac{Q_{new} - Q_{old}}{\frac{Q_{new} + Q_{old}}{2}} \cdot \frac{\frac{p_{new} + p_{old}}{2}}{p_{new} - p_{old}}. $$

You want to prove that revenue increases after a price increase, i.e. $$ Q_{new} \cdot p_{new} > Q_{old} \cdot p_{old} $$ if and only if demand is inelastic, i.e. $$ \frac{Q_{new} - Q_{old}}{\frac{Q_{new} + Q_{old}}{2}} \cdot \frac{\frac{p_{new} + p_{old}}{2}}{p_{new} - p_{old}} > -1. $$ This is fairly straightforward: multiply the inequality with the nominator, eliminate the products that appear on both sides and you have your result.

One can similarly prove that revenue decreases as price increases if and only if demand is elastic.

In case someone is curious about the point elasticy proof:

Suppose total revenue ($p \cdot D(p)$) increases as price increases, that is \begin{eqnarray*} \frac{\text{d} \left( p \cdot D(p) \right)}{\text{d} p} & > & 0 \\ \\ p \cdot \frac{\text{d} D(p)}{\text{d} p} + D(p) & > & 0. \end{eqnarray*} Rearranging this we get \begin{eqnarray*} p \cdot \frac{\text{d} D(p)}{\text{d} p} & > & - D(p) \\ \\ \frac{p}{D(p)} \cdot \frac{\text{d} D(p)}{\text{d} p} & > & - 1 \end{eqnarray*} where the expression on the left hand side is point elasticity of demand. The algebraic manipulations are reversible, thus we have shown that revenue increases as price increases if and only if demand is inelastic.

Note that here we are talking about marginal price changes; we only proved that revenue reached its local maximum. It is possible that several local maxima exist and the global maxima is elsewhere.

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