Question

Case Study 8.1

In February 2017 the price of a daily pass to drive on a Volusia County beach was $10, and at that price 26,467 daily passes were sold. In February 2018 the price of a daily pass rose to $20, and at that price the number of daily passes sold dropped to 17,994.

1. What is the elasticity of demand for the daily pass? Is the demand elastic or inelastic? Explain the meaning of your answer in the context of this problem. Is revenue increasing or decreasing?
2. The County Council may consider raising the price of a daily pass to $30 next year. Based on elasticity of demand, should the Council consider another increase? Please justify your answer.

Case Study 8.2

Demand for tickets to a theme park, based on average daily attendance, is given by           Dp=-7.7p2+495.8p+20,000, where p is the daily admission price. The current admission price is $75, but the park is considering raising the price to $80.

3. At $75, what is the elasticity of demand for the tickets? Is the demand elastic or inelastic? Based on elasticity of demand, should the price be increased?
4. What should the ticket price be for maximum revenue?

Answer 1

Question (a)

Let the number of daily passed sold be "n"

Price of each pass be "p"

For Febraruary 2017

nf2017 = 26467

pf2017 = $10

For Febraruary 2018

nf2018 = 17994

pf2018 = $20

Elasiticity of demand = % change in number of passes sold / % change in price of passes

% chage in number of passes sold = (17994 - 26467) * 100 / 26467

= -32.01%

% chage in price of passes sold = (20 - 10) * 100 / 10

= 100%

So Elasticity of Demand = -32.01% / 100%

= -0.32 which implies we have a negative price elasticity of 0.32

The elasiticity of demand is said to be elastic whe the elasticity of demand value is 1 or it is inelastic if it is <1

Here the elasticity of demand is 0.32

So the demand  is inelastic

In the context of the proble, though the price was increased by 100% from 10 to 20, the number of passed sold has not increased by 100%. So the demand has not decreased as much as the price is increased

Revenue is number of passed sold * price of each pass

For Feb 2017, the revenue is 264670 and for Feb 2018 the revenue is 359880 which implies that the revenue is increasing

Question (b)

Yes they can consider an increase in price to $30 because as we seen in the previous question though the price has increased by 100% the demand is not decreasing by only 32% approximately resulting in an 0.32 value for elasticity of demand which implies in increase of revenue

Question (c)

Given Demand = -7.7 * P2 + 495. 8 * P + 20000

for P= 75, Demand is -7.7 * 752 + 495.8 * 75 + 20000

= 13872.5

for P = 80, Demand is -7.7 * 802 + 495.8 * 80 + 20000

= 10384

% change in Demand = (10384 - 13872.5) * 100 / 13872.5

= -25.18%

% change in Price = (80 - 75) * 100 / 75

= 6.67%

Elasiticity of demand = -25.18% / 6.67% = -3.772 implies negative elasticity of demand of 3.77

since the elasticity of demand is 1, the demand is elastic

The Price should not be increased since the demand is decreasing by more % than the increase in % price which implies the revenue will decrease

Revenue at $75 price = 75 * 13872.5 = 1040438

Revenue at $80 price = 80 * 10384 = 830720

The revenue is Demand * price

So revenue is

R = (-7.7 * P2 + 495. 8 * P + 20000 ) * P

= -7.7 * P3 + 495. 8 * P2 + 20000P

We need to differentiate the equation with P and then solve that differentiated equation to arrive at the price which gives maximm revenue

Differentiating R = -7.7 * P3 + 495. 8 * P2 + 20000P with P we get

0 = - 23.1P2 + 991.6 P + 20000

We need to solve this quadratic equation

For the equation of the form ax2 + bx + c = 0, the roots are

x = (-b +- b2 - 4ac) / 2a

Here a = -23.1 , b = 991.6, c= 20000

So P values are (- 991.6 +- 991.62 - 4*(-23.1) * 20000) / 2a

= (-991.6 +- 1682.638) / -46.1

= (-991.6 + 1682.638) / -46.1 or (-991.6 - 1682.638) / -46.1

= -14.9575 or 57.88394

Since the price cannot be negative the price at which the revenue will be maximum is $57.88394