CE 561 Lecture Notes. Transportation System Performance. Set 4. -interaction between demand and supply Demand

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1 CE 561 Lecture Notes Set 4 Transportation System Performance -interaction between demand and supply Demand p, Price p 0 D 3 = α βp p 1 D 2 D 1 0 1, Vol. Short-run change in demand due to change in price Long-run shifts in demand curve Supply Quantity a producer is willing to offer at a given price p S p 0 0 Euilibrium intersection of D and S p S p* * D

2 Example: S: price = (no. of airline seats sold/day) D: seats = (price/seat) Euilibrium price = $ seats = 714 Example: tt (mins.) = volume (vehs/hr) S volume = tt D tt S Euilibrium: mins tt = mins. Vol. = 647 vehs/hr 647 veh/hr D Vol. Elasticity of Demand e = Percent change in demand due to 1% change in price or service attribute d/ d p = = Point Elasticity dp / p dp p p 0 p 1 Are Elasticity = ( 1 o )( p1 + p0 )/ 2 ( p p )( + )/ 2 1 o

3 For a linear demand function = α βp d p p e p = = β dp Substituting for p, e p α = 1 When e 1, demand is elastic Example: When the fare on a bus route was $1, the ridership was 2000/day. By reducing the fare to $0.50, the ridership increased by 600. What is the elasticity of demand with respect to fare? 600 E = 0.5 ( 1+.5)/ 2 ( ) / 2 = 0.39 < 1 inelastic Elasticity and Revenue e = % change in ridership % change in price If e > 1, demand is elastic and increase in price will reduce revenue and decrease in price will increase revenue e < 1, opposite e = 1, revenue will remain the same irrespective of the change in price

4 Demand Model = α( p) β α & β are constant parameters β is the elasticity of demand with respect to p d = αβp dp β 1 d p Ep = = αβp dp = αβp = β β 1 p β 1 1 p Model Form: the exponent of the price variable is the price elasticity Example: e p = = 12,500/day, p 0 = 0.50 p 1 = 0.75 What policy? β = αp ,500 = α( 50) α = For p = 70 cents, = 4,955 Loss of ridership = 12,500 4,955 = 7,545 Loss of revenue = 12,500 x 0.5-4,955 x 0.7 = $3,406 Example: Transit demand function = T 0.3 C 0.2 A = transit ridership/hr T = transit tt (hrs) C = transit fare ($) A = cost of auto trip ($) I = average income ($) I

5 a. 0 = 10,000/hr, p = $1 1 =?, p = 0.9 Revenue gain? e C = % reduction in fare 0.2% increase in ridership Fare reduction = (100-90)/100 = 10% Corresponding increase in ridership = 2% 1 = ,000 = 10,200/hr Rev Gain = 10,200 x ,000 x 1 = -$820/hr b. Current auto trip cost $3 (including parking). If the parking charge were raised by 30 cents, the impact on transit ridership? Auto cost cross elasticity = cents/$3 10% increase in auto cost Ridership increase 1% addl. 100 riders c. Average income of travelers $15,000/yr. What additional income necessary to offset the auto cost increase? income elasticity = % increase in income 0.25% decrease in ridership d di e I = 0.25 = / I di d =.01/ = 1% I = % di.01 = I 0.25 An increase in income of 4% (15,000 x.04 = $600) would cover a 30 cent increase (10% increase) in auto cost. That means a $600 income increase would prevent 100 riders from switching to transit.

6 Direct and Cross Elasticities Direct Elasticity the effect of change in the price of a good on the demand for the same good. Cross Elasticity the effect on the demand for a good due to a change in the price of another good. Example: A 15% increase in gasoline price has resulted in a 7% increase in bus ridership and a 9% decrease in gasoline consumption. p 0 = gasoline price before p 1 = gasoline price after 0 = gasoline consumed before 1 = gasoline consumed after Direct Elasticity: 1 = 0 x 0.91 p 1 = p 0 x 1.15 e = = p p 0.09 = ( 1+.91) ( )

7 Cross Elasticity B 0 = Bus ridership before B 1 = Bus ridership after = B 0 x 1.07 B 0.07 B ( ) e = = = 0.48 p 0.15 p ( ) Consumer Surplus difference between what consumers might be willing to pay for a service and what they actually pay. Unit Price, p p* Consumer Surplus Supply Curve Demand Curve 0 * Latent Demand Unit price, p p 1 S 1 = Existing Supply Curve S 2 = Supply Curve after improvement p = = Change in consumer surplus = System Benefit 1 2 ( p p ) + ( p p )( ) ( p )( )/ 2 p

8 Example: Urban bus service Current: capacity buses, 90% load factor fare $1 Proposed: 20% increase in fleet size 95% load factor fare $0.9 Change in consumer surplus? Revenue gain? Fare ($) Consumer Surplus 1 = 100 buses x 40 seats x 0.9 (load factor) = 3600 persons/hr Rev = 3600 x $1 = $3600/hr (persons/hr) Proposed 2 = 120 buses x 40 seats x 0.95 (load factor) = 4560 persons/hr Rev = 4560 x 0.9 = $4140/hr Change in Consumer Surplus = ( )( )/2 = $408/hr Rev Gain = ( ) = $504/hr

9 Transportation Cost A linear generalized cost function C T C T = a + bv fixed C FC variable C VC C T a Avg. Total Cost = = + b v v C FC a Avg. Fixed Cost = = v v a b v C vc Avg. Variable Cost = = b v Marginal Total Cost = dc T = b dv dc Marginal Fixed Cost = FC = 0 v dc vc Marginal Variable Cost = = b v Real world cost functions are more complex than simple linear forms. Economy of Scale Total Cost Increasing Avg. Variable Cost Law of Diminishing Marginal Productivity Production process becomes inefficient in its use of resources Output Economy of Scale decrease in AC as output increases

10 AC Economy of Scale Diseconomy of Scale Output Marginal Cost -additional cost associated with the production of an additional unit of output Unit Cost TC Unit Cost MC AC=MC AC FC 0 Q Units AC is proportional to the slope of a line connecting origin to a point on TC Q Units MC is the slope of the tangent at any point Cost Elasticity E C = % change in cost (C) % change in output () dc = C d dc d MC = = C AC

11 Example: Trucking company cost function 1.25 C = 15 C AC = = dc MC = = d 0.25 = MC = = 1.25 There is no economy of 0. AC 15 scale AC increases with EC = 25 Marginal Cost and Congestion Pricing cost/ trip C B A F G J MC AC Demand 0 L M N Beyond L, MC > AC In the absence of any interference, the flow will euilibrate at J at a cost of 0B and volume 0N If marginal cost pricing is used and a toll of FG is imposed, the flow will decrease to 0M at a cost of 0C Example: Demand Supply v = t 4 v t = Time value $5/vph Congestion toll? Highway length 10 miles

12 Time taken by all vehicles 4 v tv = v v 10 v + = 4 5 ( 2000) Marginal time ( tv) 4 d 0.75v = dv 2000 = 4 Vol. Time Demand Speed Marginal (veh/hr) (min) Flow (vph) (mph) time (min) Time (min) Demand Marginal Time Personal Time Time Reduced by Congestion Toll Volume (vph)

13 $5 Congestion Toll = ( ) 60 = $ cents/mile Optimal Flow = 2100 vph Transportation Cost Capital Agency Cost (Transp. Provider) Maintenance Operating User Cost Capital (veh purchase) Maintenance (partly variable) Operating (variable) fuel, t.t., parking, etc. Social Cost Crash Cost Air Quality Noise, etc. Highway Capital Cost Preliminary Engineering Land Acuisition (ROW) Construction Engineering Construction Facility Structures Traffic Control Devices Lighting Grading and Drainage Temp Traffic Facilities Administration & Overhead

14 Models with historical and cross sectional data Periodic Rehab Costs historical data Scoping alternative designs Residual value ROW 100% Earthwork 90% Pvt Base 80% Pvt Surface 0% Structures 50% Capital Costs for Project Planning- Major Cost Categories Typical Segments Atypical Segments Systemwide Elements Add-ons Typical Segments Constant vertical alignment nature of right-ofway, etc. Cross-section drawings Application of unit costs to per foot uantities

15 ROW for Typical Segments HOV Separate Barrier-Separated Buffer-Separated Contraflow Light Rail City Streets Separate Aerial Tunnel Cut-and-Cover Atypical Segments Right-of-way constrictions; major, uniue structures; transitions between significantly different elevations, etc. Items related to mitigation of environmental impacts System-Wide Elements Vehicles, yards, garages, etc. Station, park-and-ride lots Power and signal systems Land acuisition