1.7 — Price Elasticity — Practice Problems (Solutions)

1. Write the inverse demand function.

Simply take the demand function and solve it for p:

qD=50−0.5pqD+0.5p=500.5p=50−qDp=100−2qD

The vertical intercept (“choke price”) is $100 and the slope is −2.

2. Calculate the price elasticity of demand when the price is $10. Is this relatively elastic or inelastic?

First we need to find qD at $10.

qD=50−0.5(10)qD=50−5qD=45

Now we have the three ingredients to calculate elasticity at $10:

ϵD=1slope×pqDϵD=1−2×1045ϵD=−0.5×0.22ϵD=−0.11

The demand is relatively inelastic, as |ϵD|<1

3. Calculate the price elasticity of demand when the price is $70. Is this relatively elastic or inelastic?

First we need to find qD at $70.

qD=50−0.5(70)qD=50−35qD=15

We already have the slope (since the demand is a straight line), so now we can simply plug into the elasticity formula:

ϵD=1slope×pqDϵD=1−2×7015ϵD=−0.5×4.67ϵD≈−2.33

The demand is relatively elastic, as |ϵD|>1

4. At what price is demand unit elastic (ϵ=−1)?

ϵD=1slope×pqDFormula for elasticity−1=−0.5×pqDSet ϵD equal to −1−1=−0.5×p(50−0.5p)Plug in demand function for qD−1(50−0.5p)=−0.5pMultiply by term in parentheses0.5p−50=−0.5pDistribute the −1−50=−pAdd 0.5pp=$50Divide by −50

5. Calculate the total revenue at $10.

The total revenue is:

R=pqR=($10)(45)R=$450

6. Calculate the total revenue at $70.

The total revenue is:

R=pqR=($70)(15)R=$1,050

7. Calculate the total revenue at the price you found for question 4.

That price was p=$50. At this price, we need to find the quantity demanded. We can use the demand function:

qD=50−0.5pqD=50−0.5(50)qD=50−25qD=25

Now that we have price and quantity, revenue is:

R=pqR=($50)(25)R=$1,250

This is where revenue is maximized.