Question

The demand for pumpkins in Halloweenstan is given by: QD = 60 – 2P 1: Compute the price elasticity of demand first when the quantity of pumpkins demanded is Q1 = 20, and then when the quantity demanded is Q2 = 5. 2: Is demand inelastic or elastic at each point (that is, at Q1 = 20 and at Q2 = 5)? 3: What is the total revenue at Q1 = 20 and then at Q2 = 5, respectively? 4: Is the total revenue maximised at either Q1 = 20 or Q2 = 5? Why?

Answer 1

1. price elasticity of demand

$P_{ED}=\frac{\text{\% change in quantity}}{\text {\% change in price}}$

QD=60-2P

2P=60-QD

P=30-0.5D

Q1=20

The corresponding price is

P1=30-0.5*20

P1=20

Q2=5

The corresponding price is

P2=30-0.5*5

P2=27.5

$\text {\% change in Quantity}=\frac{20-5}{\frac{20+5}{2}}*100\%$

=120%

$\text {\% change in price}=\frac{20-27.5}{\frac{27.5+20}{2}}*100 \%$

=31.57%

$P_{ED}=\frac{120}{31.57}$

=3.8

2. elasticity of demand

The demand is elastic between the two points since elasticity is greater than 1.

3 Total revenue

total revenue= price*quantity

for Q1=20

total revenue=20*20

=400

For Q2=5

Total revenue=27.5*5

=137.5

4 total revenue maximization

revenue is maximized when the first derivative with respect to quantity is equal to 0.

R=P*Q

R=(30-0.5Q)*Q

R=30Q-0.5Q²

$R ^\prime =30-Q$

0=30-Q

Q=30

P=30-0.5Q

=30-0.5(30)

=15

Revenue at maximization point is

=30*15

=450

Total revenue is neither maximized at Q=20 nor Q=5 since the quantity at revenue maximization is 30 with a price of 15.