Why does revenue decrease with elastic demand when price increases

if a good has elastic demand and a firm increases their price, the revenue of that firm will decrease.

If demand is elastic, then the following identity must be true:

(\vartriangle \!\,Q ÷ \vartriangle \!\,P) ÷ (P ÷ Q) > 1

With some algebra, we can re-arrange it such that:

\vartriangle \!\,Q *P > \vartriangle \!\,P * Q

The term on the left hand side can be thought of as the loss of revenue, as the change in Q will be negative so you will lose how many units Q decreased by multiplied by the price.

For example, if the price is $2 and the quantity changed by 10 units, then you would have $2 * -10 = -$20. Thus your revenue decreases by $20.

The term on the right hand side can be thought of as the increase in revenue. As you are now receiving a higher price for each unit. Thus, suppose the price increased by $1 and the quantity is 15, you will now receive an extra $1 for each of those 15 units so your revenue increases by $15.

Since the term on the left is larger than then one on the right, it is the case that revenue will decrease because the "quantity effect" (the left hand side) outweighs the "price effect".

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