Bus Fares and the Revenue Question
A city transit authority currently charges $2.00 per ride and sells 500,000 rides per month, generating $1,000,000. Facing a $300,000 monthly deficit, the board proposes raising the fare to $2.50. An analyst estimates that at $2.50 ridership would fall to 440,000. Two board members disagree: one says 'demand for bus rides is inelastic, so raising the fare always helps'; the other says 'commuters will abandon us, so any fare increase loses money.' The authority also offers monthly passes, and a survey suggests that riders with passes are far less responsive to per-ride prices than cash riders. Operating costs are largely fixed: buses and drivers cost about $1,200,000 per month regardless of ridership.
1. Compute the price elasticity of demand implied by the analyst's estimate and determine whether the fare increase raises or lowers revenue. Which board member is closer to correct?
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Use the midpoint (arc) formula. Percentage change in quantity = (440,000 − 500,000) / 470,000 = −60,000 / 470,000 ≈ −12.77%. Percentage change in price = (2.50 − 2.00) / 2.25 = 0.50 / 2.25 ≈ 22.22%. Elasticity ≈ 12.77 / 22.22 ≈ 0.57, which is inelastic (|E| < 1).
Revenue at $2.50 = 440,000 × $2.50 = $1,100,000, up $100,000 from $1,000,000. So the first board member is directionally right: with inelastic demand, a price increase raises total revenue. But 'always helps' is too strong - it ignores the size of the deficit ($300,000), so the $100,000 gain covers only a third of it. The second member's claim that 'any increase loses money' is contradicted by the arithmetic. A complete answer computes the elasticity, shows the revenue gain, and notes the deficit remains $200,000, so the fare increase is necessary but insufficient.
2. Explain why pass holders and cash riders might have different elasticities, and how that affects the board's optimal pricing. Does the fixed-cost structure change the analysis?
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Elasticity depends on substitutes, budget share, and time horizon. Pass holders have already paid a sunk fee, so the marginal cost of a ride is zero to them; they ride regardless of the per-ride price, making their demand highly inelastic. Cash riders face the per-ride price at each decision and can substitute walking, cycling, or driving, so they are more elastic. The aggregate elasticity of 0.57 is a weighted average of these groups.
This suggests price discrimination: keep the pass price attractive to lock in inelastic riders and raise the per-ride cash fare more aggressively, since cash riders are the responsive group but also the group whose revenue falls least per unit price rise only if they are still inelastic. The board should estimate elasticities separately before setting a single fare.
The fixed-cost structure matters for the deficit, not for the revenue-maximizing price. With costs near $1,200,000 and revenue $1,000,000, the authority loses money at any ridership because marginal cost per rider is near zero - this is a natural-monopoly-like cost structure. Revenue maximization and deficit elimination are different targets: the board needs $1,200,000 revenue, which a $2.50 fare does not reach. It must combine pricing with subsidy or service cuts, and should not confuse 'raise revenue' with 'break even.'