Question

Revenue is maximized at what specific numerical value of the (own-)price elasticity of demand

Answer 1

Let us look at it mathematically.

Total revenue is price × quantity

In order to maximize total revenue, we need to differentiate the total revenue function with respect to the quantity twice.

And the conditions of maximization is

First order condition (FOC) =

And, second order condition (SOC) =

Now, price elasticity of demand is defined as the degree of responsiveness of quantity demanded to the change in price.

e = % change in quantity demanded / % chacha in price

Now, since total revenue is P × Q and to maximize TR, the foc must be satisfied, the change in P×Q must be equal to zero. This can only happen when the percentage change in price equals the percentage change in quantity. (Note that if price is increasing, the quantity demanded will decrease such that the product of the two (i.e. TR is constant).

When the price is very high, no (or very less) quantity is demanded and the TR = 0 (or very less) . Also, at this point the elaelastic is high because since the prices are high, the percentage change in quantity demanded will outweigh the percentage change in price. e > 1

Similarly at a point where prices are very low, the quantity demanded is very high, and the % change in price will outweigh the % change in quantity demanded. e < 1. Here too, the TR will be very low.

However, when e = 1 (unit elastic), the % change in quantity demanded equals the percentage change in price and thus the total revenue is same at both the points and hence our FOC will be satisfied and the TR will be maximized.

We just checked that for initial quantity level, as quantity rises, total revenue will rise. At some quantity it will be maximised and then with increase in quantity the total revenue will decrease. Thus, the shape of TR curve will be an inverted U and it will be maximised where the slope of the curve equals zero (that is when the total revenue is the highest evidently). This will happen when the price elasticity of demand is 1.