A ferry in Boston Harbor travels between Hingham (south of Boston) and Rowes Wharf (downtown Boston). A monthly pass on this ferry costs $329. The Massachusetts Bay Transit Authority (MBTA) is considering increasing the price for the monthly pass by 6.7%. Currently the MBTA sells 2,819 monthly passes on this route to commuters.
a. An estimate for the price sensitivity of commuters to the monthly pass price is e= -0.9. If this is the case, how many monthly passes will be sold if the price is raised by 6.7%?
b. Is the current revenue with the pass price of $329 higher or lower than the expected revenue if the price is increased by 6.7%?
c. If the -0.9 estimate is a short run elasticity, what is the expected long run number of passes sold and expected revenue if long run e= -1.2?
The Correct Answer and Explanation is:
To answer these questions, we’ll use the price elasticity of demand formula:
$$
\text{% change in quantity demanded} = e \times \text{% change in price}
$$
Where:
- $e$ = price elasticity of demand
- % change in price = 6.7% = 0.067
- Current price = \$329
- Current quantity = 2,819 passes
(a) New quantity of passes sold with price increase of 6.7% and e = -0.9
$$
\%\text{ΔQ} = -0.9 \times 0.067 = -0.0603 \text{ or } -6.03\%
$$
$$
\text{New quantity} = 2819 \times (1 – 0.0603) = 2819 \times 0.9397 \approx 2,648 \text{ passes}
$$
✅ Answer (a): About 2,648 monthly passes will be sold.
(b) Compare revenue before and after price increase
Current Revenue:
$$
\text{Revenue} = 329 \times 2819 = \$926,451
$$
New Price = $329 \times 1.067 = 351.04$
New Revenue:
$$
351.04 \times 2648 \approx \$929,548.90
$$
✅ Answer (b): New revenue (\$929,548.90) is slightly higher than current revenue (\$926,451), so expected revenue increases.
(c) Long-run scenario with e = -1.2
$$
\%\text{ΔQ (long run)} = -1.2 \times 0.067 = -0.0804 \text{ or } -8.04\%
$$
$$
\text{New quantity} = 2819 \times (1 – 0.0804) = 2819 \times 0.9196 \approx 2,593 \text{ passes}
$$
$$
\text{New revenue (long run)} = 351.04 \times 2593 \approx \$910,186.72
$$
✅ Answer (c): In the long run, about 2,593 passes will be sold, and revenue will fall to about \$910,187.
Explanation (300+ words)
This problem revolves around price elasticity of demand, which measures how sensitive consumers are to price changes. A negative elasticity (like -0.9 or -1.2) implies that as price increases, demand decreases.
In part (a), the elasticity is -0.9, and the price is increased by 6.7%. Using the elasticity formula, we calculate that the demand will fall by about 6.03%. Multiplying this decrease with the original quantity sold (2,819), we find the new estimated number of passes to be about 2,648.
In part (b), we compare total revenues before and after the price hike. Initially, the MBTA earns \$926,451 per month. After the price increase to \$351.04, even with the drop in quantity to 2,648, the new revenue slightly increases to \$929,548.90. This demonstrates that, in the short run, the demand is inelastic (|e| < 1). This means commuters don’t significantly reduce their purchases in response to price increases, so revenue rises.
In part (c), elasticity is -1.2 in the long run, meaning demand becomes more elastic (|e| > 1). Consumers have more time to adjust, possibly finding alternative transportation or remote work options. The 8.04% decline in demand leads to 2,593 passes sold, resulting in a revenue of about \$910,187—lower than both the current and short-run revenues. This indicates that long-term price hikes can backfire, especially when the good or service is more elastic over time.
In summary, while a price increase may generate more short-run revenue, the long-run response may lead to reduced total revenue if commuters become more price-sensitive. This is crucial for MBTA’s pricing decision
