Determine the toll to yield the highest revenue for part a.
A bridge has been constructed between the mainland and an island. The total cost (excluding tolls) to travel across the bridge is expressed as C = 50 + 0.5V, where V is the number of veh/hr and C is the cost/vehicle in cents. The demand for travel across the bridge is V ? 2500 ?10C .
What will be an ideal response?
Assume toll rate at T. The new cost function will be C = 50 + 0.5V + T. Since the revenue generated is the toll rate, T, time the volume, V, first solve for V with the new cost function.
V = 2500 – 10(50 + 0.5V + T)
V = 2500 – 500 – 5V – 10T
V = (2000 – 10T) / 6
Since the revenue generated is R = T × V, substitute the above expression into the revenue formula and differentiate with respect to T.
R = T × ((2000 – 10T) / 6)
R = (2000T – 10T2) / 6
dR/dT (2000T – 10T2) / 6 = 0
(2000 – 20T) / 6 = 0
Therefore, the toll which would yield the maximum revenue is T = 100, or T = $1.00.
R = T × V
R = (2000T – 10T2) / 6
R = (2000(100) – 10(100)2) / 6
R = 16,666.67
Therefore, a toll of $1.00 will yield a revenue of $166.67 per hour.
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