Download 05.0 Problem_Set_5_Elasticity

Problem Set #5
Elasticity
Introduction:
Elasticity of demand measures the percentage change in the quantity demanded caused by
a 1 percent change in either the price of the good or in the income of consumers. For
instance, the article “The Public Demand for Open Space” estimates the price and income
elasticities of the demand for open space. The equation for elasticity is:
Ep=Price Elasticity= percentage change in quantity demanded/percentage change in price.
Ei = Income Elasticity= percentage change in quantity demanded/percentage change in
Income.
We calculate the percentage change in the following way:
Percentage change in demand= Change in Demand/Quantity Demanded
Percentage change in price =Change in price/price.
Percentage change in Income =Change in Income/Income.
We expect the elasticity of demand to be negative because as the price rises the quantity
decreases. But we often ignore the minus sign and refer only to the size of the elasticity.
For example, let the demand for a good be:
P=100-10Q+5I, where I is income.
What is the demand elasticity if P= 5? We plug this value into the equation and get:
10Q=100-P+5I=95. So Q=9.5+5I/10.
Now suppose that the price increases from 5 to 6. Then the quantity becomes
Q=9.4+5I/10.
The change in the quantity demanded due to the change in price from 5 to 6 is, therefore,
a decrease in quantity demanded of 0.1. We need to calculate what percentage this
change is of the quantity demanded. But what is the quantity demanded? We have here
two quantities demanded: The first is the quantity demanded when the price is 5, the
second is when the price is 6. So we take the average of these two quantities, which is
Q=9.45+5I/10. If we assume that I=10 then the average quantity is 14.45. The
percentage change in quantity is therefore, 0.1/14.45, or .7%.
To calculate the percentage change in price we use the average of the two given prices, 5
and 6. The percentage change in price is 1/5.5=18.2%.
So an 18.2% increase in price causes a .7% decline in quantity demanded and the
elasticity is therefore:
Ep=.7/18.2=.038.
Hence, if the price were to increase by 10% the quantity demand would fall by only
.038*10%=0.38%, or less than half a percentage point. This demand is called “inelastic”
because it is relatively insensitive to changes in price. When the elasticity is less than 1
the demand is called inelastic and when it is more than 1 it is called elastic. When the
elasticity is 1 the demand is “unitary elastic.”
Problems:
1. The demand for trips per day on a toll road is P=45-.05T+.01I, where I is the median
income of the residents. The income of the residents is $1000.
The cost of producing trips is $5/trip plus a fixed cost of $xxx that is paid for by a loan
with loan payments of $5,000 a day forever.
a. What should the toll be?
b. Should the road be built?
c. Should there be any other charges? What should these other charges be?
d. What is the price elasticity at that toll?
e. What is the price elasticity when the toll is 30?
f. What is the income elasticity at the optimal toll?
2. Parts a-d as above, but the marginal cost increases with the number of trips taken
according to the equation given by MC=.005T.