CHAPTER 2
TRANSPORTATION ECONOMICS
DEFINITION
Economics is the study of how people and society choose to use scarce productive
resources to produce various commodities and distribute them among various
persons/groups in society, (e.g., guns versus butter). Economics says much about how
people behave (e.g. choice of your education). The study of economics is very important
to Transportation Engineering because it allows evaluation of alternative solutions to
transportation problems and as a result, it provides an objective means for selecting the
best alternative. The following material assumes that the basic relationships on the timevalue of money has already been covered as a part of an earlier economics class.
THE DISCOUNT RATE
The discount rate is the percentage rate which should be applied to sums of money to
have them "moved" in time, either into the past or into the future. It is often a
misunderstood quantity, being confused with the interest rate banks charge or pay. It is
conceptually the difference between the "average" interest rates banks pay and the
inflation rate. This is understood by examining the value of, say $1,000 today. They could
be placed in the bank and make, say, 5% annually. A year later they would amount to
$1,050, but they would worth less because of inflation. Say, the inflation is 3% annually.
Hence the "net" profit of putting the money in the bank is 5%-3%=2%. This is the
effective discount rate. Note that unless this difference is positive, people would not put
their $ in the bank, (e.g., Brazil's inflation rate is 2,000% per year and as result nobody
saves money in this currency). This description may be oversimplified, (i.e., what would
be the "average" bank rate one should use?), but describes the concept behind the
discount rate.
TRANSPORTATION DEMAND
It is important to understand that the transport demand is most commonly derived (i.e., it
is not demand for its own sake). That is you travel because you want to go to work or a
holiday or shopping. Only on a sunny Sunday afternoon travel demand may be self
fulfilling. The demand for travel depends on the price or cost as perceived by the traveler
for a given trip. This cost may include a number of components, (out-of-pocket, comfort,
accidents and so on).
Linear Travel Demand
Let us describe first a simple linear demand model in a linear fashion:
q = α − βp → p = ( α − q ) / β
(2.1)
where, q is the travel demand and p is the unit travel price. Note that short-term changes
in q due to p changes are along a particular line of demand. Long-term changes (such as
the ones brought by different technologies) are represented by shifts in the demand curve.
Figure 2-1.: Linear Demand Function
Let us examine the sensitivity of travel demand to changes in price, referred to as
elasticity of travel demand, ep (= percentage change in the number of trips q in response
to a 1% change in price p), expressed as:
ep =
∂q/q ∂q p
=
⋅
∂ p/ p ∂ p q
(2.2)
For a linear demand function q = α − βp , the elasticity e p becomes:
ep = − β
p
α−q
and by substituting-in p =
β
q
α−q
) = − α + 1 → ep = 1 − α
ep = − β 1 (
q
q
q
β
(2.3)
Example:
Given a linear demand function expressed by q = 200 - 10p, where q is the number of
trips taken and p is the ticket price in cents. Determine the elasticity with respect to price
ep for ticket fares of 20,15,10,5 and 0 cents. Note that from the equation above the
corresponding number of trips is 0,50,100,150 and 200.
Using Equ. 2.3, and substituting-in α = 200 , gives:
2-2
e20 = 1 − 200 = −∞
0
e15 = 1 − 200 = −3
50
e10 = 1 − 200 = −1. 000
100
e5 = 1 − 200 = −0. 333
150
e0 = 1 − 200 = 0
200
Figure 2.2: Elasticity a function level of demand (number of trips)
Observations:
• The situation where ep < -1 represents elastic demand. The resulting percentage
change in trips is larger than the percentage change in price.
•
The situation where rep ep > -1 resents inelastic demand. The resulting percentage
change in trips is smaller than the percentage change in price.
Note also that:
In the inelastic region, increases in ticket price increase the total income being
generated:
e.g., 150 x 5 = 750 ¢ or $7.5
100 x 10 = 1000 ¢ or $10
•
2-3
•
In the elastic region, increases in ticket price decrease the total revenue being
generated:
e.g., 100 x 10 = $10
50 x 15 = $7.5
A transit system operator would maximize total revenue by pricing a ticket near the upper
limit of inelasticity, i.e., at ep = -1.
Exponential Demand Model (Kraft Demand Model)
In the linear model described above, the elasticity of demand with respect to the price was
a function of the level of demand. Often, the elasticity does not dependent on the level of
demand, that is it constant. Exponential demand functions yield constant elasticities as
shown below. These models are referred to as Kraft models:
q = α pβ
(2.4)
Calculate the elasticity e p according to Equ. 2.2. It is:
∂q
= α β p β −1
∂p
(2.5)
and by substituting Equ. 2.5 into Equ. 2.2 gives:
ep = α β p
β −1
p
pβ
→ ep = α β
q
q
(2.6)
and by substituting-in the expression for Equ. 2.4 gives:
ep = α β
pβ
→ ep = β
α pβ
(2.7)
Thus, for the Kraft Model, the exponent of the demand function is the elasticity with
respect to price and it is constant, (i.e., independent of level of consumption)
Example:
The elasticity with respect to price is given equal to -2.75 (i.e., 1 unit increase in fare
results in 2.75 decrease in the number of passengers). Given that there is a ridership of
12,500 passengers at a fare of 50 ¢. Should the management raise the fare to 70¢?
Since the elasticity is constant, this implies a Kraft demand model, as:
q = αpβ with β = −2. 75 , q = αp −2.75
2-4
The known number of passengers at the given price allows "calibration" of this model.
(i.e., determination of its constant α ).
12 , 500 = α 50 −2.75 →
α = 12500 50 +2.75 = 5.876 * 108
Hence, q = 5. 876 * 10 8 p −2.75
Now that the model is calibrated, it can be used to determine the ridership for a 70¢ fare:
q = 5.876 * 108 * 70 −2.75 = 4 , 955 passengers
Hence,
•
income @ 50¢ = 12,500 x 50¢ = $6250
•
income @ 70¢ = 4,955 x 70¢ = $3468
which means that by raising the fare the company lost money. This was anticipated given
the that demand was elastic, ( ep −1).
Another Example:
In some cases, there may be need to fully calibrate a Kraft demand model, (i.e., determine
both α and β in Equ. 2.4). This is possible if 2 points are available as shown below.
p , fare c
q , passengers
12,500
5,000
50
90
Using Equ. 2.4 allows setting up 2 equations with two unknowns, α and β , as follows:
12 , 500 = α 50β
5, 000 = α 90β
Solve the system by taking logs:
→ 4. 0969 = log α + β ⋅ 1. 6990
4. 0969 = log α + β ⋅ log 50
→ 3. 6990 = log α + β ⋅ 1. 9542
3. 6990 = log α + β ⋅ log 90
Which gives:
8. 0063 = 1. 9542 og α + β 3. 3203
6. 2846 = 1. 6990 og α + β 3. 3203
1.7217 = 0.2552 ogα 
→ ogα = 6. 7465 
→ α = 5, 577 , 934
Substitute into one of earlier equations to get β :
2-5
4. 0969 = 6. 7465 + β ⋅1. 6990
β = 2. 6496 = −1. 5595
1. 6990
which suggests elastic demand, hence increases in prices would reduce total income.
Multi-Variate Exponential Demand Models (Kraft)
Usually more than price affects the number of trips for a particular transportation mode.
An example is given below of a multi-variate Kraft demand model:
q = t −0.3c −0.2 a +0.1i −0.25
Where,
q = number of trips taken by transit (Bus/LRT)
t = travel time by transit
c = travel cost by transit
a = travel cost by car
i = income
Note, that variables which cause increases in ridership have a + exponent and vice-versa.
Elasticity can be defined in a fashion similar to the one in Equ. 2.7 with respect to any of
the independent variables, (e.g., ec = elasticity with respect to cost, ei elasticity with
respect to income and so on).
Example
For the demand equation above, determine the following:
a. 10,000 pass/hr use the system at a $1.00 fare. How would the ridership change at a
$.90 fare? Would the net income of the transit company increase?
ec = −0. 2 > −1. 0 → inelastic region
For a 10% decrease in price, ($1.00 → $0.90) there will be a 0.02 increase in ridership,
hence:
10, 000 
→ 10, 200
Calculate income before and after:
10,000 x $1.0 = $10,000
10,200 x $.90 = $ 9,100
2-6
This means that the transit company will lose $820/hr. This comes by no surprise since
demand is inelastic, ( ec = −0. 2 > −1. 0 ) and hence decreases in fare are a bad idea.
b. Average trips cost $3.00 (including parking). What would be the effect on transit
ridership if there was a $.30 increase in parking cost?
ea = +0.1
This means that a unit increase in the travel cost by car will bring a 0.1 increase in transit
ridership. Hence:
10% increase
→ 0.01 transit ridership
And therefore ridership will become 10,100 pass./hr.
Elasticity for an Arbitrary Demand Function
For any form of demand function, Equ. 2.2 the derivatives can be approximated by
differences as shown below:
ep =
q 1 − q 2 ( p 1 + p2 ) / 2
p 1 − p2 ( q1 + q 2 / 2
(2.8)
Figure 2.3: Calculating Elasticity for an Arbitrary Demand Function
2-7
Consumer Surplus
It is a measure of the monetary "profit" of consumers. It is defined as the difference
between what consumers are willing to pay and the price being charged for a product or
service. This is reflected by the area (ABC) in the following figure:
Figure 2.4: Concept of the Consumer Surplus
The various attributes of the demand curve can be visualized as areas on the previous
graph:
•
•
•
Total community benefit = (AOQB)
Market Value
= (BCOQ)
Consumer surplus
= (ABC)
Supply and Demand
Usually, the supply curve, (number of units of a product a supplier is willing to produce
given a certain sale price for the product) is shown on the same graph with the demand
curve. At the intersection of the supply and demand curves the price is such that all the
units supplied are consumed by the demand. This condition represents efficiency because
there are neither units being unsold nor shortage of units for sale, (i.e., point O on Fig.
2.5).
Another way to visualize supply and demand in the transportation context is to define the
amount of travel q as the volume of vehicles per hour on a lane of road and the travel
price p as the travel time per mile, respectively. It is obvious that as vehicle volumes go
up, congestion increases and the travel time per mile goes up, (i.e., moving up on the
supply curve). On the same token, as the travel time goes down the demand for travel
2-8
Figure 2.5: Supply and Demand Curves
increases, (i.e., moving down on the demand curve). Theoretically, at the intersection of
the two curves there is enough road capacity, (number of vehicles per lane of road)
provided to satisfy all the demand with no excess capacity margin.
An improvement of a highway, (e.g. a lane widening) would move the supply curve
down, (i.e., 1 → 2 on Fig. 2.6) and would increase the consumer surplus by the amount
indicated by the shaded trapezoid.
Figure 2.6: Roadway Improvement and Consumer Surplus
The area of the trapezoid is given by:
2-9
A =
(p
1
− p2
)(q
1
+ q2
)/ 2
(2.9)
Example:
A bus company with a fleet of one hundred 40-seater buses increases its fleet size by 20%
and reduces its fare from $1.00 to $.90/ride. Calculate the change in consumer surplus
and e p .
Assumptions:
Load factor/occupancy
90% before
95% after
Before:
100 x 40 x 0.90 = 3600 p/hour
Revenue 3600 x $1.00 = $3,600/hour
After:
120 x 40 x 0.95 = 4,560 p/hour
Revenue 4,560 x $0.90 = $4,104/hour
The company gains = $4,104 -$3,600 = $504/hour
Change in consumer surplus, (Equ. 2.9) = ( p1 − p2 )(q1 + q 2 ) / 2 =(1-0.90)(3600+4560)/2
ep =
q1− q 0 ( p1 + p0 ) / 2
960 0.95
=−
ç
÷ = −2.235
.
p1 − p0 (q1 + q 0 ) / 2
010
4,080
Elastic
Therefore, even without increasing the number of buses, a reduction of fare would
increase the company's profit.
COST
Cost is typically broken down into fixed (same irrespective of level of consumption) and
variable (dependent on the number of units consumed/produced).
Total cost =Fixed Cost + Variable Cost or TC(x) =FC + VC(x)
(2.10)
An important concept to understand is that of marginal cost. It can be defined as the
total cost for producing an additional unit of output or change of total cost per additional
unit produced. This can be expressed as:
MC(x) = TC(x) - TC(x-1)
(2.11)
2-10
Or if total cost is a continuous function, marginal cost could be expressed as the
derivative of total cost, as follows:
MC ( x ) =
∂ TC ( x )
∂ VC ( x )
∂
→ MC ( x ) =
FC + VC ( x ) → MC ( x ) =
∂x
∂x
∂x
(2.12)
Example.
The cost of running a train system with a variable number of wagons is given in the
following table:
# Wagons/Train Fixed
1
2
3
4
5
6
7
8
9
10
55
55
55
55
55
55
55
55
55
55
Var
30
55
75
105
155
225
315
425
555
705
Total
Average
Marginal
85
110
130
160
210
280
370
480
610
760
85.0
55.0
43.3
40.0
42.0
46.7
52.9
60.0
67.8
76.0
--25
20
30
50
70
90
100
130
150
Demonstrate the relationship between the cost attributes by plotting them against the
number of wagons, (Figure 2.7).
Points to be made:
•
•
•
The slope of the total cost curve is equal to the marginal cost curve, (e.g., it has a
∂ TC ( x )
minimum for about 2 wagons). Mathematical explanation: MC ( x ) =
∂x
The average cost is proportional to the slope of the line defined by the origin and any
point along the total cost curve. Explanation: the slope is equal to the total cost
divided by the number of wagons, (i.e., average).
The tangent shown represents lowest average cost
•
The Marginal Cost Curve (MC) and the Average Cost Curve (AC) intersect at a
point which coincides to the lowest average cost. Explanation: because the tangent of
total cost curve from origin in average cost coincides with the derivative of the total cost
(i.e., marginal cost) at that point.
2-11
Figure 2.7: Total, Average and Marginal Costs, (Khisty 1991).
2-12
In mathematical terms this is:
min AC ( x ) = ∂ AC ( x ) / ∂ x = 0 → ∂ ( TC ( x ) / x ) / ∂ x = 0 →
∂ ( TC ( x ). x −1 ) / ∂ x = 0 → ∂ TC ( x ) / ∂ x x −1 − TC ( x ). x −2 = 0 →
∂ TC ( x ) / ∂ x = TC ( x ) / x → MC ( x ) = AC ( x )
Hence, lowest total cost is achieved at the intersection of MC and AC curves.
Laws related to cost
•
Law of Diminishing Returns: Increase in input of one factor of production may
cause an increase in output. Eventually, a point will be reached beyond which
increasing units of input will cause progressively less increase in output.
e.g. utility of the 11th beer
•
Law of Increasing Return to Scale or Economics of Mass Production: The
situation in which increasing all factors of production brings about a larger
increase in output.
e.g. Mac Donald's fast-food
The main reason for this is that fixed costs are divided among more production
units.
Example:
A car costs $9,000 and will last for 10 years if used 12,000 miles/year.
Fuel, tires, maintenance cost is $.10 per mile.
Tax and insurance is $500/year.
Car depreciates 1% every 1,200 miles (life of 120,000 miles).
Discount rate of 10%
Determine the annual average cost and the cost/mile.
Annual Fixed Cost
•
Depreciation:
2-13
i (1 + i )
$9,000 x( A / P,10%,10 years) = 9000
(1 + i ) n −1
n
→ $9000 x 0.1627 =
9000
= $1, 464. 7
6.1446
• Insurance and taxes:
$500
Therefore, Total Fixed Cost (FC)= $1,464.7+$500=$1,964.7
Variable Cost (VC):
•
Fuel, tires, maintenance=12,000 x $0.10 = $1,200/year
Total Cost:
Total Annual Cost = FC + VC = $1964.70 + $1,200 = $3,164.70/year
AC($/ mile) =
$3164. 70
= $0. 2637 / mile
12, 000
Pricing and Subsidy Policies
All of you are familiar with the problems of traffic congestion in metropolitan areas. To
alleviate traffic congestion, a roadway authority must decide on the action to be taken.
The available alternatives are:
•
•
•
Do nothing
Add road capacity, (more traffic lanes on main arterials)
Restrict/tax private auto usage
The first two alternatives have been shown to result in more congestion in the long run.
The third alternative incorporates 3 broad choices:
•
•
•
Tax suburban/dispersed living
Subsidize public transportation
Road pricing or other methods to discourage private auto usage for auto trips to the
Central Business District (CBD). (e.g., Singapore)
The following issue arises: If auto trips are to be taxed, what is the level of taxation that
optimizes utilization of the transportation system? To address this problem, let us look at
the supply-demand curves in terms of:
2-14
p = cost/trip, generalized travel cost, (=travel time+vehicle operating costs)
q = traffic flow volumes in vehicles per hour (vph)
Let us plot, average cost (AC), marginal cost (MC) and the demand curve (DD)on a
graph:
Figure 2.8: Equilibrium (J) and Optimum Traffic Conditions (F) on a Roadway, (Khisty
1991).
The short-term cost to motorists is given by the AC curve. At low traffic volumes, the
cost per trip remains unchanged. In the absence of an external interference, the vehicle
flow will equalize at a level J, where the average cost (AC) curve and the demand curves
(DD) intersect, (i.e., this reflects the condition whereby the vehicle flow is such that
results in a travel cost/time which the average driver is willing to tolerate).
As vehicle volumes increase above point J, however, additional vehicles delay the entire
traffic stream causing dramatic increases in total travel cost/time. It has hence been
observed that the optimum traffic condition is achieved at the intersection of the marginal
cost curve (MC) with the demand curve (DD), (i.e., point F where the total increases in
travel time from the addition of 1 vehicle are so high as to discourage that vehicle from
entering the traffic stream). This can achieved by levying a tax equal to GF,
($/trip/vehicle) thus artificially raising the average cost to bring about a reduction of
traffic volumes from N to M. The benefit is a reduction in the travel cost/time of all
remaining vehicles, while the loss is the lost benefit from the MN trips. This is consistent
with what economists call marginal cost pricing and can be stated as "a unit increase in
total cost brings about a unit decrease in demand".
Example:
The Federal Highway Administration (FHWA) suggests a general form of travel time
curves as a function of traffic volume. For traversing, say 10 miles, the travel time t is:
2-15
4
é
V
t = 10ê1 + 015
. ç
÷ ú
2,000 ú
êë
(2.14)
where, V = traffic flow in vph
t = travel time in minutes
This implies that the vehicle volumes above 2,000 vph (termed the capacity of the road)
have a 4th power impact on travel time. Let's define the demand function as d=4000-100t
If highway users value their travel time @ $5/hour what is the congestion toll that should
be levied?
Set up a table to facilitate drawing AC and MC curves. Marginal cost is derived by:
V5
Vt = 10êV + 015
.
2,000 4 ú
∂ Vt
V4
= 10ê1 + 0.75
ú
∂ V
2,000 4
or
t m arg inal
V4
= 10ê1 + 0.75
ú
2,000 4
Tabulate cost figures for traffic volumes of 1,000, 2,000 and so on vph:
vph
t(min)
Demand
Speed Marginal Time
∂ = 4000 − 100t
(mph)
(min)
___________________________________________________________
1000
1500
2000
2500
3000
10.09
10.47
11.50
13.66
17.59
2991
2953
2850
2634
1241
59.46
57.31
52.17
43.92
21.75
2-16
10.5
12.4
17.5
28.3
48.0
Figure 2.9: Example of Determining Optimum Traffic Volume and Toll , (Khisty 1991).
Graphically the following curve intersections can be determined:
•
•
AC and Demand curve at (2560,14.3)
MC and Demand curves at (2120,18.8)
Toll: 18.8-14.3 = 4.5 minute and translate into $ by:
4. 5 x
$5
= 37. 5cents
60
Hence the optimum flow is 2120 vph and it can be produced by charging $0.37 per trip
per vehicle.
2-17