Price elasticity of demand (PED)
Price elasticity of demand (PED) shows
the relationship between price and quantity demanded and provides a
precise calculation of the effect of a change in price on quantity
demanded. The following equation enables PED to be calculated.
We can use this equation to calculate the
effect of price changes on quantity demanded, and on the
revenue received by firms before and after any
price change.
For example, if the price of a daily newspaper
increases from £1.00 to £1.20p, and the daily sales falls from 500,000 to
250,000, the PED will be:

50% + 20%
= () 2.5
The negative sign indicates that P and Q are inversely related, which we would expect for most price/demand relationships. This is significant because the newspaper supplier can calculate or estimate how revenue will be affected by the change in price. In this case, revenue at £1.00 is £500,000 (£1 x 500,000) but falls to £300,000 after the price rise (£1.20 x 250,000).
The range of responses
The degree of response of quantity
demanded to a change in price can vary considerably. The key
benchmark for measuring elasticity is whether the coefficient is
greater or less than proportionate. If quantity demanded changes
proportionately, then the value of PED is 1, which is called ‘unit
elasticity’.
PED can also be:

Less than one, which means PED is inelastic.

Greater than one, which is elastic .

Zero (0), which is perfectly inelastic.

Infinite (∞), which is perfectly elastic.
PED along a linear demand
curve
PED on a linear demand curve will fall
continuously as the curve slopes downwards, moving from left to
right. PED = 1 at the midpoint of a linear demand curve.
PED and revenue
There is a precise mathematical connection between
PED and a firm’s revenue.
There are three ‘types’ of revenue:

Total revenue (TR), which is found by multiplying price by quantity sold (P x Q).

Average revenue (AR), which is found by dividing total revenue by quantity sold (TR/Q).

Marginal revenue (MR), which is defined as the revenue from selling one extra unit. This is calculated by finding the change in TR from selling one more unit.
Revenue Consider theses figures and calculate Total, Marginal and Average Revenue. 

When TR is at a maximum, MR =
zero, and PED = 1.

Price and AR are identical, because AR = TR/Q, which is P x Q/Q, and cancel out the Qs to get P.

A curve plotting AR (=P) against Q is also a firm’s demand curve.

TR increases, reaches a peak and decreases.

When TR is at a maximum, MR is zero.

MR falls at twice the rate of AR.
Why does a firm want to know PED?
There are several reasons why firms gather
information about the PED of its products. A firm will know much more
about its internal operations and product costs than it will about its
external environment. Therefore, gathering data on how consumers respond
to changes in price can help reduce risk and uncertainly. More
specifically, knowledge of PED can help the firm forecast its sales and
set its price.
Sales forecasting
The firm can forecast the impact of a change
in price on its sales volume, and sales revenue (total revenue, TR). For
example, if PED for a product is () 2, a 10% reduction in price (say,
from £10 to £9) will lead to a 20% increase in sales (say from 1000 to
1200). In this case, revenue will rise from £10,000 to £10,800.
Pricing policy
Knowing PED helps the firm decide whether to raise
or lower price, or whether to price discriminate. Price discrimination is a policy of
charging consumers different prices for the same product. If demand is elastic, revenue is gained by
reducing price, but if demand is inelastic, revenue is
gained by raising price.
Nonpricing policy
When PED is highly elastic, the firm can use
advertising and other promotional techniques to reduce elasticity.
Determinants of PED
There are several reasons why consumers may
respond elastically or inelastically to a price change, including:
The number and ‘closeness’ of substitutes
A unique and desirable product is likely to
exhibit an inelastic demand with respect to price.
The degree of necessity of the good
A necessity like bread will be demanded
inelastically with respect to price.
Whether the good is habit forming
Consumers are also relatively insensitive to changes in
the price of habitually demanded products.
The proportion of consumer income which is spent on
the good
The PED for a daily newspaper is likely to be much lower
than that for a new car!
Whether consumers are loyal to the brand
Brand loyalty reduces sensitivity to price changes
and reduces PED.
Life cycle of product
PED will vary according to where the product is in
its life cycle. When new products are launched, there are often
very few competitors and PED is relatively inelastic. As other firms
launch similar products, the wider choice increases PED. Finally, as a
product begins to decline in its lifecycle, consumers can become very
responsive to price, hence discounting is extremely common.
The effects of advertising
Firms may use persuasive advertising by to win new
customers and retain the loyalty of existing ones.
Advertisers use a range of media, including
television, press, and electronic media. Advertising will shift demand
to the right, and make demand less elastic.
There are three extreme cases of PED.

Perfectly elastic, where only one price can be charged.

Perfectly inelastic, where only one quantity will be purchased.

Unit elasticity, where all the possible price and quantity combinations are of the same value. The resultant curve is called a rectangular hyperbola.
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