Question

The per-week (inverse) demand for use of the Øresund Bridge between Denmark and Sweden is P = 13 − 0.15Q during peak traffic periods and P = 10 − 0.1Q during off-peak hours, where Q is the number of cars crossing the bridge in thousands and P is the toll in euros. If the marginal cost of using the bridge is MC = 5 + 0.2Q, what is the optimal peak load toll and off-peak load toll for crossing the bridge?

Answer 1

Demand during Peak $P = 13−0.15Q$

                $MC = 5+0.2Q$

Since,    $TR =P×Q$

 $TR =(13−0.15Q)Q$

          $TR = 13Q−0.15Q^2$

           $MR = \frac{∆TR}{∆Q}$

$MR = 13−0.30Q$

At optimal level     MC = MR

         $5+0.2Q = 13−0.30Q$

              $0.50Q = 8$

$Q=\frac{8}{0.50}$

$= 16$

$P =13−0.15(16)$

$=13−2.4 = 10.6$

Optimal peak load toll for crossing the bridge is 10.6

Demand during off-Peak load $P = 10−0.1Q$

$MC = 5+0.2Q$

Since,                $TR =P×Q$

 

                     $TR =(10−0.1Q)Q$

$TR = 10Q−0.1Q^ 2$

$MR=\frac{\Delta TR}{\Delta Q}$

$=10-0.20Q$

At optimal level,       MC = MR

               $5+0.2Q = 10−0.20Q$

              $0.40Q = 5\\Q=\frac{5}{0.40}=12.5$

$P =10−0.1(12.5) =10−1.25 =8.75$

Optimal off-peak load toll for crossing the bridge is 8.75