On the same demand curve, PAX (000s) = 320 − 0.4 × Fare (MEL–LST), at what fare is demand exactly unit elastic, and what does this imply for total passenger revenue? Which of the following statements is TRUE?

Question

BlackTom AI · Accepted Answer

For a linear demand Q = a − bP, point price elasticity ε = (dQ/dP)·(P/Q) = −bP/(a − bP). Unit elasticity (|ε| = 1) therefore satisfies bP/(a − bP) = 1.

1) Relevant knowledge points and concepts
- Demand is linear: Q = a − bP. Here Q (in thousands) = 320 − 0.4·Fare, so a = 320 and b = 0.4.
- Price elasticity of demand for linear demand: ε = −bP/(a − bP).
- Total revenue TR = P·Q. TR is maximized where marginal change dTR/dP = 0, which occurs where ε = −1 (unit elasticity).

2) Step-by-step reasoning and algebra
Set bP/(a − bP) = 1:
bP = a − bP
2bP = a
P = a/(2b)

3) Intermediate calculations with the given numbers
a = 320, b = 0.4:
P = 320 / (2·0.4) = 320 / 0.8 = 400 (AUD)

Compute Q at P = 400:
Q = 320 − 0.4·400 = 320 − 160 = 160 (thousands of passengers) = 160,000 pax

Compute total passenger revenue:
TR = P · Q = 400 AUD × 160,000 pax = 64,000,000 AUD = AUD 64.0M

4) Why this is correct
Unit elasticity (|ε| = 1) for a linear demand occurs at the midpoint and is the maximum of the TR curve because dTR/dP = Q + P·dQ/dP = Q(1 + ε). Setting this equal to zero gives ε = −1, so TR is maximized at the unit-elastic price. Thus P = AUD 400 yields the maximum TR = AUD 64.0M.

5) Why other options are incorrect
- (a) P = AUD 800 gives Q = 0, TR = 0, but 800 is not the unit-elastic point (elasticity is undefined at Q = 0).
- (b) Correct price and TR but incorrectly calls it the minimum; it is a maximum.
- (d) TR value 32.0M is incorrect arithmetic (400 × 80,000 would give 32M, but Q is 160,000, not 80,000).

6) Conclusion
The correct statement is: Unit-elastic fare = AUD 400, giving total revenue of AUD 64.0M, the maximum point on the total revenue curve (choice c). This matches the provided answer.

On the same demand curve, PAX (000s) = 320 − 0.4 × Fare (MEL–LST), at what fare is demand exactly unit elastic, and what does this imply for total passenger revenue? Which of the following statements is TRUE?单项选择题

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