Efficient Road Prices

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Efficient Road Prices

• Economics of Public PolicyPADM 625
• Ben Muse

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Congestible public good

• Non-rival when there are not too many consumers
• But becomes congested as the number of consumers
  gets larger

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Bridges and roads

• Provide an example
• When traffic is light, a highway or a bridge may
  approximate a non-rival state
• Drivers dont interfere with each others use
• When traffic is heavy, they may approximate a
  rival state
• One driver use may also perfectly displace
  another driver

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What about excludability?

• Roads may be excludable or non-excludable
• Toll systems may be relatively inexpensive in
  rural areas with few access points
• Yet prohibitively expensive in urban areas with
  lots of access points

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Congestibility

• Use of the facility may change by day or time of
  day
• If facility is excludable at a reasonable price
• Efficiency may mean charging a price that varies
  with the number of drivers

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91 ExpressLanes

• Private firm
• Operates private toll road
• On the median of a state highway
• (competition with state highway prevents
  monopoly)

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Speed and congestion

• Relate average speed to the volume of traffic
• He uses the following formula
• S is average speed in miles/hour
• N is number of cars (in hundreds)
• S60 for N lt 5
• S300/N for N gt 5

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Speed and congestion

• So, if there are 1000 cars on the road,
• N 10
• And S 300/10 30 miles per hour

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How long to travel a mile?

• Speed in miles per hour is S300/N
• The time it takes to travel one mile is hours
  per mile or 1/S
• So we can relate the time to travel a mile to the
  number of cars like this
• 1/S N/300
• So if S 60 mph, 1/S 1/60 or a minute

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Average cost to travel a mile

• The average time cost to travel a mile
• Will be equal to the product of
• The value of an hour of the drivers time
• And the time it takes to travel a mile
• AC W(1/S) W(N/300)

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Average cost

• So
• If the drivers time is 10/hour
• And there are 400 drivers on the road
• So that the average speed is 60 mph
• The average cost per mile would be
• AC 10(1/60) 0.167

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Average cost

• Note that, above 500 cars
• The AC increases by 3 1/3 cents for every
  additional 100 cars

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Graphing average cost
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Marginal cost

• Marginal cost is very important
• Because we are going to want to add cars to the
  road
• Until the marginal cost of an additional car
• Is equal to the marginal benefit

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Marginal cost

• Average cost
• AC W(N/300)
• Total cost
• Is equal to average cost per motorist times the
  number of motorists
• TC NW(N/300)
• Or TC W(N2/300)

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Marginal cost

• The marginal cost of an extra driver is equal to
  the first derivative of total cost with respect
  to the number of drivers, N
• TC W(N2/300)
• MC 2W(N/300)
• MC (2WN)/300 2AC

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MC and AC

• According to the formula,
• MC 2AC
• So, if AC 5 cents, MC 10 cents
• If AC 10 cents, MC 20 cents
• Note this only holds above 500 cars

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Adding marginal cost
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Marginal benefit curve

• Now we need to look at the benefit side
• And add in a marginal willingness to pay curve
  (MWTP)

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Traffic with no toll

• With no toll, people will operate until their own
  marginal benefit is equal to their cost (given by
  the average cost curve)
• But these people will be ignoring the costs
  imposed on others by their activity (represented
  by the marginal cost curve)

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Adding marginal cost
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The optimal traffic level

• Equates the marginal benefit a person receives by
  traveling another mile
• With the cost to them of traveling that mile,
  plus
• The cost they impose on others by traveling that
  mile

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Optimal toll and traffic
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So,

• The optimal toll
• Will be equal to the distance between the average
  cost curve and the marginal cost curve
• And internalizes for the driver the costs
  imposed on others

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MWTP curve may change by day and hour
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The optimal toll

• Varies by time of day
• When the MWTP curve is far to the right, demand
  is heavy
• The costs imposed on others by ones driving are
  higher
• And the optimal toll is higher

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The optimal toll

• At night
• The MWTP curve is far to the left
• And, if it is positioned as shown
• The optimal toll is zero
• No congestion driving a mile is non-rival, a
  positive toll would cut off valuable driving with
  no gain.

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Route 91 users

• No toll booths
• Members set up prepaid toll accounts
• Have transponders attached to their cars
• Overhead antennas and computers read
  transponders, calculate toll for the lane, day,
  time of day
• Accounts are billed

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Sources

• http//www.91expresslanes.com/learnaboutus.asp?pm
  7 accessed on September 15, 2003.
• Bruce, Chapter 3, Public Goods in Theory and
  Practice