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10. Elasticities
Essay
1.
What is price elasticity of demand? Describe the three ranges of elasticity and their relationship to marginal
revenue.
Problems
2.
3.
Given appropriate information, calculate the arc price, income, and cross-price elasticities of demand and
identify.
Given appropriate information, calculate the point price, income, and cross-price elasticities of demand and
identify.
Price Elasticity of Demand
When a firm is thinking about setting its price, the responsiveness of quantity demanded to price is very
important. The law of demand states that if the relative price of a good rises, the quantity demanded falls. But by
how much? If the amount buyers are able and willing to buy is very responsive to changes in the good’s price, then
buyers will purchase much less. If it isn’t very responsive, then they will buy about the same amount—only a little
bit less.
The term used in economics to refer to this responsiveness is elasticity.
Price elasticity of demand is the percent change in quantity demanded divided by the percent change in
the good’s price.
There are two measures of elasticity.
Arc price elasticity of demand Ep= Q2-Q1/P2-P1 * P2+P1/Q2+Q1
Point Price elasticity of demand Ep = first derivative of the demand function * P/Q
Qd=65060-20P
derivative = -20
point price elasticity of demand function = -20P/Q
Suppose the price is 2753. Using the demand function, quantity is 10,000. The point price elasticity of demand is:
Ep = -20 * P/Q
= -20 * 2753/10,000
= -5.51
If you are given two prices and the associated quantities demanded, then the arc elasticity of demand can be
calculated:
P
500
550
Qd
249,500
248,500
Ep = (249,500 – 248,500)/(500-550) * (500+550)/(249500+248500)
= 1000/-50 * 1050/498,000
= -20 * .002108
= -.04217
If you are just given two prices and a demand function, arc elasticity of demand can still be found. First,
however, you must substitute the two prices into the demand function and find the quantity demanded.
Ranges of Elasticity
Inelastic-elasticity is less than one (in absolute value)
Inelastic means that a change in price causes a less than proportional change in quantity demanded. For
example, a 1% increase in price causes a less than one percent decrease in quantity demanded.
If demand is inelastic then a higher price raises revenue even though less is sold. And a lower price means less
revenue, even though more units are sold.
If demand is inelastic, then marginal revenue is negative. That means an increase in output results in lower
revenue and a decrease in quantity leads to more revenue. That’s because a bit more output goes with a lower price
and even though more units are sold, the lower price for each unit causes less revenue.
Elastic-elasticity is greater than one in absolute value.
Elastic means that a change in price causes a more than proportional change in quantity demanded. For
example, a 1% increase in price causes a more than 1% decrease in quantity demanded.
If demand is elastic, then a higher price lowers revenue and a lower price raises revenue. Even though more is
earned for each unit when the price rises, the number of units sold drops so much that revenue is less. And a lower
price raises revenue because so many more units are sold even though less is made per unit.
If demand is elastic, marginal revenue is positive. A larger output is associated with more revenue and a
smaller output with less revenue.
Unit Elastic - elasticity is equal to one in absolute value.
Unit elastic means that a change in price causes a proportionate change in quantity demanded. For example, a
1% increase in price leads to a 1% decrease in quantity demanded.
If demand is unit elastic, then a higher or lower price leaves revenue unchanged.
An increase in price
generates more revenue per unit, but the proportionate decrease in units sold leaves revenue unchanged.
If demand is unit elastic, then marginal revenue is equal to zero. If quantity increases, there are more units
sold, but the lower price per unit leaves revenue unchanged.
Perfectly Inelastic
Elasticity is equal to zero. This implies that a change in price has no impact on quantity demanded. It is
inconsistent with the law of demand.
Perfectly Elastic
Elasticity is infinite. An increase in price, no matter how small, causes quantity demanded to fall to zero. A
decrease in price, no matter how small, causes quantity demanded to rise to infinity. While literally not possible, a
small firm in a large market selling a product identical to those of its competitors might perceive itself in this
position. Any increase in price above what the others’ charge will result in no sales. Lowering the price would not
literally result in infinite sales, but charging less than the going price will result in more orders than can possibly be
filled.
Income Elasticities
Income elasticity of demand looks at the responsiveness of demand to changes in income. It is the percent
change in quantity over the percent change in income.
Arc income elasticity
Arc income elasticity of demand is calculated when given two different levels of income and the quantity demanded
at each income level.
Ey= Qd2-Qd1/y2-y1 * y2+y1/Qd2+Qd1
Example:
Y
7100
7200
Qd
249500
270000
Epy = (249500-270000)/(7100-7200) * (7100+7200)/(249500+270000)
= 5.64
Point Income Elasticity
Point income elasticity of demand is calculated when there is a demand function.
Ey = first derivative of the demand function with respect to income * y/Q
Qd = -20,000 - 20P + .5A^.5 - 50Pc +.1Ps + 5Y
derivative = 5
point income elasticity = 5y/Q (plug in the value of income and quantity demanded.)
Ey = 5*9000/10,000
= 4.5
income elasticity greater than zero--normal good
income elasticity less than zero--inferior good
income elasticity greater than one--luxury
income elasticity less than one--necessity
Generally, economists define a normal good as one with a negative income elasticity of demand. If
income rises, then quantity demanded falls and that’s it. While we say that such goods are lower quality
and give those sorts of example, the defining characteristic is what happens to demand when income
changes.
Similarly, in income elasticity of demand is positive, then the good is normal, regardless of ones
opinion about its quality.
In economics, a luxury is defined as a good with an income elasticity of demand greater than one. That
means that a change in income leads to a more than proportional change in quantity demanded. If income
rises, quantity demanded rises more than in proportion. If income falls, quantity demanded falls more than
in proportion.
In economics, a necessity is a good with income elasticity of demand less than one (but greater than
zero. Inferior goods aren’t generally considered necessities.) If income should rise, demand for a
necessity rises, but less than in proportion to the increase in income. And vice versa.
Cross Price Elasticity of Demand
Cross-price elasticity demand looks at the responsiveness of the demand for a good to the price of some
other good. It is the percent change in quantity of the good divided by the percent change in quantity of
some other good s price. The formulas look similar to ordinary price elasticity of demand-Arc elasticity Ep= Q2-Q1/Po2-Po1 * Po2+Po1/Q2+Q1
point elasticity Ep = first derivative of the demand function with respect to some other good * Po/Q
Again, this must be done using the demand function, not the simple one.
Qd = -20,000 - 20P + .5A^.5 - 50Pc +.1Ps + 5Y
Let s do Pc
derivative = -50
point cross price elasticity = -50Pc /Q (plug in the value of the other goods price and quantity demanded of
the good we are working with.)
Epc = -50*6/10000
= -.03
cross price elasticity greater than zero, the goods are substitutes
cross price elasticity less than zero, the goods are complements.
Again, economists define substitute and complement according to these relationships. If the increase in
another goods price leads to an increase in the demand for the good, then they are substitutes. While the
rationale is that somehow people can use one good instead of another, there is no need to see some obvious
way in which people might do this. While we use examples like rice crispies and corn flakes, it doesn’t
have to be that obvious.
Complements are the same. If an increase in another goods price leads to a lower demand for the
good, then they are complements. The idea is that the goods are somehow used together. And while we
give examples like milk and cereal, no such obvious relationship is necessary—just the relationship
between the price and quantity demanded.