Mathematics?

Posted on 2016-06-17 18:44:00 in Math by MILO

Suppose the state wants to try and reduce traffic congestion by imposing tolls on drivers. To figure out the socially optimal toll, make the following assumptions: 
value of a driver’s time not spent in a car is $30/hour. The number of cars on the road is denoted N, and the overall traffic speed in miles per hour is denoted S. When there are less than 5 cars on the road (N<5), traffic moves at 60 MPH. When there are 5 or more cars (N≥5), the speed is S=300/N.a. What is the average cost per mile (AC), in terms of the number of cars on the road? Calculate this for N<5 and N≥5. Graph the AC curve, with N on the x-axis, $ on the y-axis. 
[One can show that for N<5, the marginal cost MC=AC. For N≥5, MC=N/5. Graph this on the same graph as part a.] 
b. During the evening, the marginal benefit for drivers is MB=6-(N/10). 
i. What is the market equilibrium number of drivers in the evening? 
ii. What is the socially efficient equilibrium number of drivers in the evening? 
iii. What, if any, is the optimal toll level?

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