Two Fare Proposals, One Elasticity, and a Claim That Fails Arithmetic
A regional transit authority carries 4.0 million trips a year at a flat $2.00 fare, generating $8.0 million in fare revenue against $14.0 million in operating cost. Staff estimate the short-run own-price elasticity of demand at about -0.30 and propose to treat it as constant across the range under consideration. The board has two proposals. Proposal A raises the fare to $2.50 to narrow the operating gap. Proposal B cuts the fare to $1.50 and pairs the cut with a $1.0 million grant, on the theory that added ridership will "pay for itself." A board member asserts that both proposals raise revenue, because demand is inelastic. A separate statutory goal requires the authority to maximize ridership without increasing the net public subsidy.
1. Compute ridership, fare revenue, and net public subsidy under the status quo and under each proposal.
Read the worked answer
Status quo
- Ridership 4.00M; revenue $8.00M; net subsidy = $14.0M - $8.0M = $6.00M.
Proposal A: fare to $2.50 (+25%)
- Ridership change: 25% x (-0.30) = -7.5%, so 4.00M x 0.925 = 3.70M trips.
- Revenue: $2.50 x 3.70M = $9.25M (up $1.25M).
- Net subsidy: $14.0M - $9.25M = $4.75M (down $1.25M).
Proposal B: fare to $1.50 (-25%), plus $1.0M grant
- Ridership change: (-25%) x (-0.30) = +7.5%, so 4.00M x 1.075 = 4.30M trips.
- Revenue: $1.50 x 4.30M = $6.45M (down $1.55M).
- Net public support required: $14.0M - $6.45M - $1.0M = $5.55M ... except the grant is a subsidy, so count it: total public money = $5.55M in direct subsidy plus $1.0M grant = $6.55M, up $0.55M on the status quo.
Summary table
| Scenario | Ridership | Fare revenue | Net public cost |
|---|---|---|---|
| Status quo | 4.00M | $8.00M | $6.00M |
| A ($2.50) | 3.70M | $9.25M | $4.75M |
| B ($1.50 + $1.0M) | 4.30M | $6.45M | $6.55M |
2. Evaluate the board member's claim that inelastic demand means both proposals raise revenue.
Read the worked answer
The claim is half right and therefore misleading.
Inelastic demand (|e| < 1) means a price increase raises revenue - that part is correct, and Proposal A demonstrates it: a 25% price rise costs 7.5% of riders and yields $1.25M more.
The error is transferring that result to a price cut. Inelasticity predicts the opposite direction: cutting price by 25% costs 7.5% of revenue in volume terms and recovers only 7.5% in riders, so revenue falls - here by $1.55M. A price cut raises revenue only if demand is elastic (|e| > 1). The board member has applied a rule about increases to a decrease.
Second error: the "pays for itself" model. Added ridership cannot pay for itself at a lower fare, by construction: each added rider contributes $1.50 against a marginal cost that is positive. The $1.0M grant does not close the $1.55M revenue loss; net public cost rises by $0.55M.
Price of the ridership gain: $0.55M for 300,000 additional trips ≈ $1.83 per added trip. That is the number the board should debate - it may be a good buy, but it is a purchase, not a self-financing investment.
3. Which proposal, if either, satisfies the statutory goal, and what would you recommend to the board? Include the limits of the elasticity estimate.
Read the worked answer
Neither proposal satisfies the goal as stated.
- A raises revenue and cuts net subsidy but reduces ridership by 300,000 - it fails the ridership half of the goal.
- B raises ridership by 300,000 but raises net public cost by $0.55M - it fails the subsidy half.
The goal as written is a constrained optimization with no feasible point among the two options, so the honest advice is to say so rather than pick the lesser violation.
Recommendation: reject the binary. The goal becomes tractable if the fare is segmented rather than flat:
- Peak pricing. Raise fares on the congested peak, where elasticity is lowest and where added riders would be least welcome; leave off-peak and weekend fares alone or reduce them, capturing ridership where capacity exists.
- Targeted discounts. Means-tested or employer passes buy ridership from the most responsive segment at a fraction of a system-wide cut.
- Cost side. At $14.0M cost against 4.0M trips, cost per trip is $3.50 against a $2.00 fare - the gap is not purely a pricing problem.
Limits of the -0.30 estimate: it is short-run, so riders adjust less now than they will in two years; a constant-elasticity assumption applied to a 25% fare change is a linearization that typically overstates ridership response to a cut and understates attrition from a large increase. Peak and off-peak elasticities differ substantially, and the estimate is an average that hides that. Any recommendation should carry a sensitivity band, not a point forecast.