# PRICE ELASTICITY AND REVENUE

|We have defined demand to be elastic when the absolute value of the price elasticity is greater than one. For that to be true, the percentage change in quantity must be greater than the percentage change in price (% change in Q > % change in P). If this were true, what would you expect to happen to a firm’s receipts if the price were lowered? Recall from principles of economics that total revenue (TR) is equal to price (P) times quantity (Q).

Consider an extreme case. Suppose that a five-percent cut in price stimulates a fifty-per cent increase in sales (the price elasticity would be 10). You would expect revenues to rise. The relatively small drop in price would be more than compensated for by a large increase in sales. To see exactly the relationship between total revenue and price elasticity, let’s return to the demand function given by the equation **Q = 400 – 4P** which is used as the basis for Table 2. The table shows how the price elasticity of demand varies along the demand curve.

*Note that no point elasticity can be calculated when Q = 0 because division by zero is not defined.*

In this table, total revenue and marginal revenue are included, as well as the point price elasticities. Marginal revenue (MR) is defined as the rate of change in total revenue, or the additional revenue generated by selling one more unit. In this example, the demand function can be solved for P in terms of Q as follows:

Consider Q = 400 – 4P

Or 4P = 400 – Q

Or P = 100 – 0.25Q

Multiplying by Q and taking the first derivative yields:

TR = P.Q

TR = (100 – .25Q)Q

TR = 100Q – 0.25Q^{2}

MR = dTR/dQ

MR = 100 – 0.5Q** **

The total revenue and marginal revenue functions along with the demand curve, are plotted in Figure 1. Notice that the slope of the marginal revenue function is twice the slope of the demand function. You see in Figure-1 that demand is price elastic over the range of quantities for which marginal revenue is positive. Because marginal revenue is the slope of total revenue (remember that **MR = dTR/dQ**), you can tell that increasing sales by lowering price will cause total revenue to rise over this interval. However, lowering price when demand is inelastic (beyond Q = 200) will result in reduced revenues. From an examination of Figure -1 and Table-2 we can reach some important conclusions about the relationship between elasticity and total revenue.

As is evident from the above discussion, the change in expenditure when price changes is related to the elasticity of demand. If elasticity is less than unity (inelastic), the percentage change in price can exceed the percentage change in quantity. The price change will then be the dominant one of the two changes and the revenue will change in the same direction as the price change. If however, elasticity exceeds unity (elastic), the percentage change in quantity will exceed the percentage change in price. The percentage change in quantity will be the more important change, so that total expenditure will change in the opposite direction as the price change. These results can be summarized as follows:

*Elastic Demand*

- Decrease price……. Increase total revenue
- Increase price………Decrease total revenue

Price and total revenue move in opposite directions.* *

*Inelastic Demand*

- Decrease price……. Decrease total revenue
- Increase price………Increase total revenue

Price and total revenue move in the same direction.

If a demand function has a **unitary elasticity**, then the same level of revenue will be generated, regardless of price. You see that for a linear demand function, as price falls, demand becomes less elastic or more inelastic. You have also seen that when demand is elastic, price cuts are associated with increases in total revenue. But if price continues to be lowered in the range in which demand is inelastic, total revenue will fall. **Thus, total revenue will be maximized at the price (and related quantity) at which demand is unitarily elastic. **You can see from Figure-1 that this point is also where MR = 0. This observation makes sense because marginal revenue is the slope of total revenue. Recall that any function may have a maximum where its slope is zero. For total revenue, we have

TR = P.Q = (100 – 0.25Q) Q

TR = 100Q – 0.25Q^{2}

dTR/dQ = 100 – 0.5Q

Setting the first derivative equal to 0 and solving for Q, we find

dTR/dQ = 100 – .5Q = 0

–.50Q = –100

Q = 200

Checking the second-order condition, we see that

d^{2}TR/dQ^{2} = – 0.5 < 0

Because the second derivative is negative, we know that total revenue is a maximum at Q = 200. The price at which 200 units will sell is Rs. 50, so that is the revenue-maximizing price.

The relationship between elasticity and total revenue can also be shown using calculus. Total revenue is price times quantity. Taking the derivative of total revenue with respect to quantity yields marginal revenue:

The equation states that the additional revenue resulting from the sale of one more unit of a good or service is equal to the selling price of the last unit (P), adjusted for the reduced revenue from all other units sold at a lower price (QdP/dQ). This equation can be written as

Because marginal revenue is zero, a price change would have no effect on total revenue. In contrast, if demand is elastic, say e_{p} = –2, marginal revenue will be greater than zero. This implies that a price reduction, by stimulating a considerable increase in demand would increase total revenue. This equation also implies that if demand is inelastic, say e_{P} = – 0.5, marginal revenue is negative, indicating that a price reduction would decrease total revenue.

Some analysts question the usefulness of elasticity estimates. They argue that elasticities are redundant, such that the data necessary for their determination could be used to determine total revenues directly. Thus managers could assess the effects of a change in price without knowledge of price elasticity. Although this is true, elasticity estimates are valuable to the extent that they provide a quick way of evaluating pricing policies. For example, if demand is known to be elastic, it is also known that a price increase will reduce total revenues. Likewise, if the Finance Minister wants to raise revenue through taxation he knows that increasing duties on cigarettes will most likely generate the result he desires.