The total operating revenues of a public transportation authority are $100 million, while its total operating costs are $120 million. The price of a ride is $1, and the price elasticity of demand for public transportation has been estimated to be -0.4. By law, the public transportation authority must take steps to eliminate its operating deficit.
The financial predicament faced by the public transportation authority illustrates a classic scenario of economic inefficiency and constrained policy options. With revenues at $100 million against costs of $120 million, the authority is operating at a $20 million deficit. The core challenge is designing an effective pricing strategy that addresses this deficit, considering the demand elasticity and the constraints imposed by law and market responsiveness.
Understanding the demand elasticity is crucial. The price elasticity of demand (PED) measures the responsiveness of quantity demanded to changes in price. A PED of -0.4 implies that a 1% increase in price would lead to a 0.4% decrease in quantity demanded. The negative sign indicates the typical inverse relationship between price and demand. Given this, any price adjustment will influence revenue and ridership levels, thus impacting the subsidy gap.
The primary goal is to close the $20 million deficit without exacerbating the decline in ridership or violating legal constraints. Since the current revenue is insufficient, and demand is relatively inelastic (-0.4), increasing prices could be a viable option. The inelastic demand suggests that revenue could potentially increase with a price hike because the proportional decrease in quantity demanded would be less than proportional increase in price, leading to higher total revenue.
Applying the concept of point elasticity, the optimal price can be derived from the marginal revenue formula. For inelastic demand, raising prices tends to increase total revenue, which is preferable in closing the deficit. Therefore, the authority should consider implementing a fare increase, carefully calibrated to maximize revenue without causing a drastic drop in ridership.
However, it is essential to consider public policy constraints and social equity concerns. A modest fare increase could be justified, especially if funds are reinvested to improve service quality or maintain service
levels. Alternatively, implementing targeted subsidies or fare discounts for low-income riders could balance revenue objectives with social responsibilities.
If the authority cannot reduce costs and needs to eliminate the $20 million deficit through fare adjustments alone, the necessary fare increase can be calculated using demand elasticity principles. It involves increasing the current fare from $1 to a new price that compensates for the revenue shortfall.
The initial revenue is $100 million at a $1 fare, with ridership being R. Therefore, R = 100 million / 1 = 100 million rides annually. To generate an additional $20 million in revenue, the total revenue must be increased to $120 million.
Let P be the new fare. The total revenue (TR) after the price increase is TR = P × R(P), where R(P) is the new ridership at price P. Using the elasticity formula:
Percentage change in quantity demanded = Elasticity × Percentage change in price
Mathematically:
R(P) = R■ × (P / P■)^(elasticity), where R■ = 100 million, P■ = $1, and elasticity = -0.4.
Rearranged, the total revenue becomes:
TR = P × R■ × (P / P■)^(elasticity) = P × 100 million × (P / 1)^{ -0.4 }
We want TR = $120 million, so:
$120 million = P × 100 million × P^{ -0.4 }
Dividing both sides by 100 million:
1.2 = P × P^{ -0.4 } = P^{ 1 - 0.4 } = P^{ 0.6 }
Solving for P:
P^{ 0.6 } = 1.2
P = (1.2)^{1 / 0.6} = (1.2)^{1.6667} ≈ 1.2^{1.6667} ≈ 1.38
Thus, the public transportation authority should increase the fare from $1 to approximately $1.38 per ride
to offset the $20 million deficit.
This fare increase results in a new revenue of approximately $138 million if ridership declines accordingly, thus eliminating the deficit under the demand elasticity assumptions. However, it’s vital to recognize that a significant fare hike might impact ridership, even with inelastic demand, and the authority should consider gradual adjustments and community impact assessments.
Conclusion
In conclusion, given the inelastic demand for public transportation, increasing fares appears to be the most feasible way to eliminate the operating deficit without reducing costs. A carefully calibrated fare hike from $1 to approximately $1.38 per ride could generate the necessary additional revenue. Nonetheless, policymakers should strike a balance between revenue objectives, social equity, and ridership retention. Complementary measures such as targeted subsidies or investment in service quality could enhance the sustainability of the transportation network while fulfilling legal obligations.
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