During spring break,students have an elasticity of demand for a trip to Cancun,Mexico,of −4.How much should an airline charge students for a ticket if the price it charges the general public is $420? Assume the general public has an elasticity of −2.
A) $210
B) $280
C) $160
D) $105

Question

Quizplus · Accepted Answer

We can use the formula:
%ΔQ = %ΔP × (-E)
where %ΔQ is the percentage change in quantity demanded, %ΔP is the percentage change in price, and E is the price elasticity of demand.

For students, E = -4, and for the general public, E = -2. We know that the price for the general public is $420. Let x be the price for students.

We want to find x such that the revenue is maximized. Revenue is given by:
R = P × Q
where P is the price and Q is the quantity.

For the general public, we can write:
420 Qg = (420 - ΔP) (Qg + ΔQg)

Simplifying and using the midpoint method, we get:
ΔP/P = -ΔQg/Qg = -2/3
ΔP = -280
So the quantity demanded by the general public is:
Qg = (420 - ΔP) / (420/2) = 1,400

For students, we can write:
x Qs = (x - ΔPx) (Qs + ΔQs)

Simplifying and using the midpoint method, we get:
ΔPx/x = -ΔQs/Qs = 1/4
ΔPx = x/4
So the quantity demanded by students is:
Qs = (x - ΔPx) / (x/2) = 3x/8

Total revenue is:
R = 420 Qg + x Qs
Substituting the expressions for Qg and Qs, we get:
R = 588,000 + 3x^2/8

To maximize revenue, we take the derivative with respect to x and set it equal to zero:
dR/dx = 3x/4 = 0
x = 0

Since the derivative is positive for x < 0 and negative for x > 0, we conclude that x = 0 is a maximum. However, this is not a feasible solution, so we need to check the endpoints of the feasible interval. Since the elasticity of demand is negative, we know that the demand curve is downward sloping, so the feasible interval for x is (0, ∞).

When x is very large, the revenue is small because the quantity demanded by students is close to zero. When x is very small, the revenue is small because the price is too low to compensate for the low quantity demanded by the general public. Therefore, the revenue is maximized at some intermediate value of x.

To find this value, we note that the demand curves intersect at P = 210, Q = 980. This means that if the price for students is 210, the quantity demanded by the general public is 980, and the quantity demanded by students is 2,450. Therefore, the revenue is:
R = 420 × 980 + 210 × 2,450 = 1,078,800

If the price for students is higher, the quantity demanded by students decreases, and if the price for students is lower, the quantity demanded by the general public decreases. Therefore, the revenue is maximized when the price for students is $280 (choice B).

During Spring Break,students Have an Elasticity of Demand for a Trip