# Transportation Economics/Print version – Wikibooks, open books for an open world

Transportation moves people and goods from one place to another using a variety of vehicles across different infrastructure systems. It does this using not only technology (namely vehicles, energy, and infrastructure), but also people’s time and eﬀort; producing not only the desired outputs of passenger trips and freight shipments, but also adverse outcomes such as air pollution, noise, congestion, crashes, injuries, and fatalities.

Figure 1 illustrates the inputs, outputs, and outcomes of transportation. In the upper left are traditional inputs (infrastructure (including pavements, bridges, etc.), labor required to produce transportation, land consumed by infrastructure, energy inputs, and vehicles). Infrastructure is the traditional preserve of civil engineering, while vehicles are anchored in mechanical engineering. Energy, to the extent it is powering existing vehicles is a mechanical engineering question, but the design of systems to reduce or minimize energy consumption require thinking beyond traditional disciplinary boundaries.

On the top of the ﬁgure are Information, Operations, and Management, and Travelers’ Time and Eﬀort. Transportation systems serve people, and are created by people, both the system owners and operators, who run, manage, and maintain the system and travelers who use it. Travelers’ time depends both on freeﬂow time, which is a product of the infrastructure design and on delay due to congestion, which is an interaction of system capacity and its use. On the upper right side of the ﬁgure are the adverse outcomes of transportation, in particular its negative externalities:

All of these factors are increasingly being recognized as costs of transportation, but the most notable are the environmental eﬀects, particularly with concerns about global climate change. The bottom of the ﬁgure shows the outputs of transportation. Transportation is central to economic activity and to people’s lives, it enables them to engage in work, attend school, shop for food and other goods, and participate in all of the activities that comprise human existence. More transportation, by increasing accessibility to more destinations, enables people to better meet their personal objectives, but entails higher costs both individually and socially. While the “transportation problem” is often posed in terms of congestion, that delay is but one cost of a system that has many costs and even more benefits. Further, by changing accessibility, transportation gives shape to the development of land.

Transportation is a process of production as well as being a factor input in the production function of firms, cities, states and the country. Transportation is produced from various services and is used in conjunction with other inputs to produce goods and services in the economy. Transportation is an intermediate good and as such has a “derived demand”. Production theory can guide our thinking concerning how to produce transportation efficiently and how to use transportation efficiently to produce other goods.

More broadly, one has transportation as an input into a production process. For example, the Gross National Product (GNP) of the economy is a measure of output and is produced with capital, labor, energy, materials and transportation as inputs.
GNP = f(K, L, E, M, T)

Alternatively we can view transportation as an output: e.g. passenger-miles of air service, ton-miles of freight service or bus-miles of transit service. These outputs are produced with inputs including transportation.

We will focus on the latter view in this chapter.

Production processes involve very large numbers of inputs and outputs. It is usually necessary to aggregate these in order to keep the analysis manageable; examples would include types of labor and types of transportation.

In transportation, output is a “service” rather than product. It is not storable (capacity unused now cannot be sold later, this leads to the economics of peak/off-peak) and users participate in the production (passengers are key elements in producing the output).

Production is characterized by multidimensional (heterogeneous) outputs.

Examples of the use of the production approach for system design considering both inputs and outputs are illustrated in the following table:

Lumpy investments refer to indivisibility of investments leads to complex costing and pricing. E.g you cannot build half a lane or half a runway and have it be useful.

Sunk investments can constitute an entry barrier.

Joint production occurs when it is unavoidable to produce multiple outputs in fixed proportions, e.g. fronthaul-backhaul problem; there is a joint cost allocation problem. Joint costs are where the multiple products are in fixed invariant proportions.

In common production, multiple outputs of varying proportions are produced using same equipment or facility – cost saving benefits, e.g. freight and passenger services using a same airplane, or using a same train. Common costs are where multiple services can be produced in variable proportions for the same cost outlay

Carriers have a structure that can be decomposed into two primary activities (Terminal and Linehaul)

Terminal activities include loading, unloading and sorting of goods (and, perhaps, pick up and delivery). The concept of speed can be important for terminals, while distance to be travelled is only of limited relevance. Terminal activities may differ depending upon the type of cargo., e.g. we see increasing returns to scale for bulk loading facilities, while it is not clear whether or not there are increasing returns to scale for facilities handing diverse product types.

Linehaul activities exhibit indivisibility of output unit on the supply side due to:

Theory of production analyzes how a firm, given the given technology, transform its inputs (

${displaystyle x}$

) into outputs (

${displaystyle y}$

) in an economically efficient manner. A production function,

${displaystyle y=f(x)}$

, is used to describe the relationship between outputs and inputs.

X-Efficiency is the effectiveness with which a given set of inputs are used to produce outputs. If a firm is producing the maximum output it can given the resources it employs, it is X-efficient.

Allocative efficiency is the market condition whereby resources are allocated in a way that maximizes the net benefit attained through their use. In a market under this condition it is impossible for an individual to be made better off without making another individual worse off.

Technical efficiency refers to the ability to produce a given output with the least amount of inputs or equivalently, to operate on the production frontier rather than interior to it.

The Production Possibilities Set is the set of feasible combinations of inputs and outputs. To produce a given number of passenger trips, for example, planes can refuel often and thus carry less fuel or refuel less often ands carry more fuel. Output is vehicle trips, inputs are fuel and labor.

If the production possibilities set (PPS) is convex, it is possible to identify an optimal input combination based on a single condition. However, if the PPS is not convex the criteria becomes ambiguous. We need to see the entire isoquant to find the optimum but without convexity we can be ‘myopic’, as illustrated on the right.

${displaystyle Cleft(Q,2P_{1},2P_{2}right)=2Cleft(Q,P_{1},P_{2}right)}$

${displaystyle {frac {partial C}{partial Q_{j}}}>0forall j}$

${displaystyle {frac {partial C}{partial P_{j}}}=Xleft(bullet right)}$

As input prices rise we always substitute away from the relatively more expensive input.

${displaystyle {frac {partial ^{2}C}{partial P_{i}^{2}}}leq 0forall i}$

Production functions are relationships between inputs and outputs given some technology. A change in technology can affect the production function in two ways. First, it can alter the level of output because it affects all inputs and, second, it can increase output by changing the mix of inputs. Most production functions are estimated with an assumption of technology held constant. This is akin to the assumption of constant or unchanging consumer preferences in the estimation of demand relationships.

The functional form represents the inputs are combined. These can range from a simple linear or log-linear (Cobb-Douglas) relationship to a the second order approximation represented by the ‘translog’ function.

Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus. For example if

${displaystyle alpha =0.15}$

, a 1% increase in labor would lead to approximately a 0.15% increase in output.

the production function has constant returns to scale. That is, if L and K are each increased by 20%, Y increases by 20%. If

returns to scale are increasing. Assuming perfect competition and

${displaystyle alpha +beta =1}$

,

${displaystyle alpha }$

and

${displaystyle beta }$

can be shown to be labor and capital’s share of output.

The translog production function is a generalization of the Cobb–Douglas production function. The name translog stands for ‘transcendental logarithmic’.

where L = labor, K = capital, and Y = product.

Constant elasticity of substitution (CES) function:

${displaystyle Y=A[alpha K^{gamma }+(1-alpha )L^{gamma }]^{frac {1}{gamma }}}$

${displaystyle gamma =0}$

corresponds to a Cobb–Douglas function,

${displaystyle Y=AK^{alpha }L^{1-alpha }}$

The Leontief production function applies to situations in which inputs must be used in fixed proportions; starting from those proportions, if usage of one input is increased without another being increased, output will not change. This production function is given by

The examination of production relationships requires an understanding of the properties of production functions. Consider the general production function which relates output to two inputs (two inputs are used only for exposition and the conclusions do not change if more inputs or outputs are considered, it is simply messier)

Consider fixing the amount of capital at some level and examine the change in output when additional amounts of labor (variable factor) is added. We are interested in the

${displaystyle Delta Q/Delta L}$

which is defined as the marginal product of labor and the

${displaystyle Q/L}$

the average product of labor. One can define these for any input and labor is simply being used as an example.

This is a representation of a ‘garden variety’ production function. This depicts a short run relationship. It is short run because at least one input is held fixed. The investigation of the behavior of output as one input is varied is instructive.

Note that average product (AP) rises and reaches a maximum where the slope of the ray,

${displaystyle Q/L}$

is at a maximum and then diminishes asymptotically.

Marginal product (MP) rises (area of rising marginal productivity), above AP, and reaches a maximum. It decreases ( area of decreasing marginal productivity) and intersects AP at AP’s maximum . MP reaches zero when total product (TP) reaches a maximum. It should be clear why the use of AP as a measure of productivity (a measure used very frequently by government, industry, engineers etc.) is highly suspect. For example, beyond

${displaystyle MP=0}$

,

${displaystyle AP>0}$

${displaystyle Q=f(K,L)}$

.

${displaystyle dQ={frac {partial f}{partial K}}dK+{frac {partial f}{partial L}}dL=0}$

rearranging one can see that the ratio of the marginal productivities (

${displaystyle {frac {MP_{K}}{MP_{L}}}}$

) equals

${displaystyle {frac {dk}{dL}}}$

Equivalently, the isoquant is the locus of combinations of K and L which will yield the same level of output and the slope (

${displaystyle {frac {dk}{dL}}}$

) of the isoquant is equal to the ratio of marginal products.

The ratio of MP’s is also termed the “marginal rate of technical substitution ” MRTS.

As one moves outward from the origin the level of output rises but unlike indifference curves, the isoquants are cardinally measurable. The distance between them will reflect the characteristics of the production technology.

The isoquant model can be used to illustrate the solution of finding the least cost way of producing a given output or, equivalently, the most output from a given budget. The innermost budget line corresponds to the input prices which intersect with the budget line and the optimal quantities are the coordinates of the point of intersection of optimal cost with the budget line. The solution can be an interior or corner solution as illustrated in the diagrams below.

The method of Lagrange Multipliers is a method of turning a constrained problem into an unconstrained problem by introducing additional decision variables. These ‘new’ decision variables have an interesting economic interpretation.

{displaystyle {begin{aligned}&{text{Max }}gleft({bar {x}}right)\&{text{s}}{text{.t}}{text{. }}h_{j}left({bar {x}}right)=b_{j}\end{aligned}}}

${displaystyle {text{Max}}Lambda left({bar {x}},{bar {lambda }}right)=gleft({bar {x}}right)-sum {{bar {lambda }}_{j}left(h_{j}left({bar {x}}right)-b_{j}right)}}$

${displaystyle {frac {partial Lambda }{partial x_{i}}}={frac {partial g}{partial x_{i}}}-sum limits _{j}{lambda _{j}}{frac {partial h_{j}}{partial x_{i}}}=0}$

${displaystyle {frac {partial Lambda }{partial lambda _{j}}}=-h_{j}left(xright)+b_{j}=0}$

Lagrange multipliers represent the amount by which the objective function would change if there were a change in the constraint. Thus, for example, when used with a production function, the Lagrangian would have the interpretation of the ‘shadow price’ of the budget constraint, or the amount by which output could be increased if the budget were increased by one unit, or equivalently, the marginal cost of increasing the output by a unit.

First order conditions (FOC) are not sufficient to define a minimum or maximum.

The second order conditions are required as well. If, however, the production set is convex and the input cost function is linear, the FOC are sufficient to define the maximum output or the minimum cost.

A profit maximizing firm will hire factors up to that point at which their contribution to revenue is equal to their contribution to costs. The isoquant is useful to illustrate this point.

Consider a profit maximizing firm and its decision to select the optimal mix of factors.

${displaystyle {frac {partial Pi }{partial K}}=P{frac {partial f}{partial K}}-r=0}$

${displaystyle {frac {partial Pi }{partial L}}=P{frac {partial f}{partial L}}-w=0}$

This illustrates that a profit maximizing firm will hire factors until the amount they add to revenue [marginal revenue product] or the price of the product times the MP of the factor is equal to the cost which they add to the firm. This solution can be illustrated with the use of the isoquant diagram.

The equilibrium point, the optimal mix of inputs, is that point at which the rate at which the firm can trade one input for another which is dictated by the technology, is just equal to the rate at which the market allows you to trade one factor for another which is given by the relative wage rates. This equilibrium point, should be anticipated as equivalent to a point on the cost function. Note that this is, in principle, the same as utility pace and output space in demand. It also sets out an important factor which can influence costs; that is, whether you are on the expansion path or not.

In order to move from production to cost functions we need to find the input cost minimizing combinations of inputs to produce a given output. This we have seen is the expansion path. Therefore, to move from production to cost requires three relationships:

The ‘production cost function’ is the lowest cost at which it is possible to produce a given output.

There is a duality between the production function and cost function. This means that all the information contained in the production function is also contained in the cost function and vice-versa. Therefore, just as it was possible to recover the preference mapping from the information on consumer expenditures it is possible to recover the production function from the cost function.

Suppose we know the cost function C(Q,P’) where P” is the vector of input prices. If we let the output and input prices take the values C˚, P˚1 and P˚2, we can derive the production function.

1. Knowing specific values for output level and input prices means that we know the optimal input combinations since the slope of the isoquant is equal to the ratio of relative prices.

2. Knowing the slope of the isoquant we know the slope of the budget line

3. We know the output level.

We can therefore generate statements like this for any values of Q and P’s that we want and can therefore draw the complete map of isoquants except at input combinations which are not optimal.

One important concept which comes out of the production analysis is that the demand for a factor is a derived demand; that is, it is not wanted for itself but rather for what it will produce. The demand function for a factor is developed from its marginal product curve, in fact, the factor demand curve is that portion of the marginal product curve lying below the AP curve. As more of a factor is used the MP will decline and hence move one down the factor demand function. If the price of the product which the factor is used to produce the factor demand function will shift. Similarly technological change will cause the MP curve to shift.

Recall that our production function Q = f(x1, x2) can be translated into a cost function so we move from input space to dollar space. the production function is a technical relationship whereas the cost function includes not only technology but also optimizing behavior.

The translation requires a budget constraint or prices for inputs. There will be feasible non-optimal combinations of inputs which yield a given output and a feasible-optimal combination of inputs which yield an optimal solution.

Technical change can enter the production function in essentially three forms; secular, innovation and facility or infrastructure.

Technical change can affect all factors in the production function and thus be ‘factor neutral’ or it may affect factors differentially in which case it would be ‘factor biased’.

The consequence of technical change is to shift the production function up (or equivalently, as we shall see, the cost function down), it can also change the shape of the production function because it may alter the factor mix.

This can be represented in an isoquant diagram as indicated on the right.

If relative factor prices do not change, the technical change may not result in a new expansion path, if the technical change is factor neutral, and hence it simply shifts the production function up parallel. If the technical change is not factor neutral, the isoquant will change shape, since the marginal products of factors will have changed, and hence a new expansion path will emerge.

First order conditions (FOC) are not sufficient to define a minimum or maximum.

The second order conditions are required as well. If, however, the production set is convex and the input cost function is linear, the FOC are sufficient to define the maximum output or the minimum cost.

Costs

## Introduction

Price, cost and investment issues in transportation garner intense interest. This is certainly to be expected from a sector that has been subject to continued public intervention since the ninteenth century. While arguments of market failure, where the private sector would not provide the socially optimal amount of transportation service, have previously been used to justify the economic regulations which characterized the airline, bus, trucking, and rail industries, it is now generally agreed, and supported by empirical evidence, that the move to a deregulated system, in which the structure and conduct of the different modes are a result of the interplay of market forces occurring within and between modes, will result in greater efficiency and service.

Many factors have led to a reexamination of where, and in which mode, transportation investments should take place. First, and perhaps most importantly, is the general move to place traditional government activities in a market setting. The privatization and corporatization of roadways and parts of the aviation systems are good examples of this phenomenon. Second, there is now a continual and increasing fiscal pressure exerted on all parts of the economy as the nation reduces the proportion of the economy’s resources which are appropriated by government. Third, there is increasing pressure to fully reflect the environmental, noise, congestion, and safety costs in prices paid by transportation system users. Finally, there is an avid interest in the prospect of new modes like high speed rail (HSR) to relieve airport congestion and improve in environmental quality. Such a major investment decision ought not be made without understanding the full cost implications of a technology or investment compared to alternatives.

This chapter introduces cost concepts, and evidence on internal costs. The chapter on Negative externalities reviews external costs.

In imperfectly competitive markets, there is no one-to-one relation between P and Q supplied, i.e., no supply curve. Each firm makes supply quantity decision which maximises profit, taking into account the nature of competition (more on this in pricing section).

Supply function (curve). specifies the relationship between price and output supplied in the market. In a perfectly competitive market, the supply curve is well defined.
Much of the work in transportation supply does not estimate Supply-curve. Instead, focus is on studying behaviour of the aggregate costs (in relation to outputs) and to devising the procedure for estimating costs for specific services (or traffic). Transport economists normally call the former as aggregate costing and the latter as disaggregate costing. For aggregate costing, all of the cost concepts developed in micro-economics can be directly applied.

## Types of Costs

There are many types of costs. Key terms and brief definitions are below.

### Shared costs

The production of transport services in most modes involves joint and common costs. A joint cost occurs when the production of one good inevitably results in the production of another good in some fixed proportion. For example, consider a rail line running only from point A to point B. The movement of a train from A to B will result in a return movement from B to A. Since the trip from A to B inevitably results in the costs of the return trip, joint costs arise. Some of the costs are not traceable to the production of a specific trip, so it is not possible to fully allocate all costs nor to identify separate marginal costs for each of the joint products. For example, it is not possible to identify a marginal cost for an i to j trip and a separate marginal cost for a j to i trip. Only the marginal cost of the round trip, what is produced, is identifiable.

Common costs arise when the facilities used to produce one transport service are also used to produce other transport services (e.g. when track or terminals used to produce freight services are also used for passenger services). The production of a unit of freight transportation does not, however, automatically lead to the production of passenger services. Thus, unlike joint costs, the use of transport facilities to produce one good does not inevitably lead to the production of some other transport service since output proportions can be varied. The question arises whether or not the presence of joint and common costs will prevent the market mechanism from generating efficient prices. Substantial literature in transport economics (Mohring, 1976; Button, 1982; Kahn, 1970) has clearly shown that conditions of joint, common or non-allocable costs will not preclude economically efficient pricing.

• Traceable cost (untraceable cost): A cost which can (cannot) be directly assigned to a particular output (service) on a cause-and-effect basis. Traceable (untraceable) costs may be fixed or variable (or indivisible variable). Traceability is associated with production of more than one output, while untraceable costs possess either (or both) common costs and joint costs. The ability to identify costs with an aggregate measure of output supplied (e.g. the costs of a round trip journey) does not imply that the costs are traceable to specific services provided.
• Joint cost: A cost which is incurred simultaneously during the production for two or more products, where it is not possible to separate the contributions between beneficiaries. These may be fixed or variable (e.g. cow hides and cow steaks).
• Common cost: A cost which is incurred simultaneously for a whole organization, where it cannot be allocated directly to any particular product. These may be fixed or variable (e.g. the farm’s driveway).

### External and Internal Costs

External costs are discussed more in Negative externalities

Economics has a long tradition of distinguishing those costs which are fully internalized by economic agents (internal or private costs) and those which are not (external or social costs). The difference comes from the way that economics views the series of interrelated markets. Agents (individuals, households, firms and governments) in these markets interact by buying and selling goods are services, as inputs to and outputs from production. A firm pays an individual for labor services performed and that individual pays the grocery store for the food purchased and the grocery store pays the utility for the electricity and heat it uses in the store. Through these market transactions, the cost of providing the good or service in each case is reflected in the price which one agent pays to another. As long as these prices reflect all costs, markets will provide the required, desirable, and economically efficient amount of the good or service in question.

The interaction of economic agents, the costs and benefits they convey or impose on one another are fully reflected in the prices which are charged. However, when the actions of one economic agent alter the environment of another economic agent, there is an externality. An action by which one consumers purchase changes the prices paid by another is dubbed a pecuniary externality and is not analyzed here further; rather it is the non-pecuniary externalities with which we are concerned. More formally, “an externality refers to a commodity bundle that is supplied by an economic agent to another economic agent in the absence of any related economic transaction between the agents” (Spulber, 1989). [1] Note that this definition requires that there not be any transaction or negotiation between either of the two agents. The essential distinction which is made is harm committed between strangers which is an external cost and harm committed between parties in an economic transaction which is an internal cost. A factory which emits smoke forcing nearby residents to clean their clothes, cars and windows more often, and using real resources to do so, is generating an externality or, if we return to our example above, the grocery store is generating an externality if it generates a lot of garbage in the surrounding area, forcing nearby residents to spend time and money cleaning their yards and street.

There are alternative solutions proposed for the mitigation of these externalities. One is to use pricing to internalize the externalities; that is, including the cost which the externalities impose in the price of the product/service which generate them. If in fact the store charged its customers a fee and this fee was used to pay for the cleanup we can say the externality of ‘unsightly garbage’ has been internalized. Closer to our research focus, an automobile user inflicts a pollution externality on others when the car emits smoke and noxious gases from its tailpipe, or a jet aircraft generates a noise externality as it flies its landing approach over communities near the airport. However, without property rights to the commodities of clean air or quiet, it is difficult to imagine the formation of markets. The individual demand for commodities is not clearly defined unless commodities are owned and have transferable property rights. It is generally argued that property rights will arise when it is economic for those affected by externalities to internalize the externalities. These two issues are important elements to this research since the implicit assumption is that pricing any of the externalities is desirable. Secondly, we assume that the property rights for clean air, safety and quiet rest with the community not auto, rail and air users. Finally, we are assuming that pricing, meaning the exchange of property rights, is possible. These issues are considered in greater detail in Chapter 3 where the broad range of estimates for the costs of the externalities are considered.

### Other terms

• Sunk costs: These are costs that were incurred in the past. Sunk costs are irrelevant for decisions, because they cannot be changed.
• Indivisible costs: Do not vary continuously with different levels of output or must expenditures, but be made in discrete “lumps”. Indivisible costs are usually variable for larger but not for smaller changes in output
• Escapable costs (or Avoidable costs): A cost which can be avoided by curtailing production. There are both escapable fixed costs and escapable variable costs. The escapability of costs depends on the time horizon and indivisibility of the costs, and on the opportunity costs of assets in question.

## Time Horizon

Once having established the cost function it must be developed in a way which makes it amenable to decision-making. First, it is important to consider the length of the planning horizon and how many degrees of freedom we have. For example, a trucking firm facing a new rail subsidy policy will operate on different variables in the short run or a period in which it cannot adjust all of its decision variables than it would over the long run, the period over which it can adjust everything.

Long run costs, using the standard economic definition, are all variable; there are no fixed costs. However, in the short run, the ability to vary costs in response to changing output levels and mixes differs among the various modes of transportation. Since some inputs are fixed, short run average cost is likely to continue to fall as more output is produced until full capacity utilization is reached. Another potential source of cost economies in transportation are economies of traffic density; unit cost per passenger-kilometer decreases as traffic flows increase over a fixed network. Density economies are a result of using a network more efficiently. The potential for density economies will depend upon the configuration of the network. Carriers in some modes, such as air, have reorganized their network, in part, to realize these economies.

In the long run, additional investment is needed to increase capacity and/or other fixed inputs. The long run average cost curve, however, is formed by the envelope of the short run average cost curves. For some industries, the long run average cost often decreases over a broad range of output as firm size (both output and capacity) expands. This is called economies of scale. The presence of economies at the relevant range of firm size means that the larger the size of the firm, the lower the per-unit cost of output. These economies of scale may potentially take a variety of forms in transportation services and may be thought to vary significantly according to the mode of transportation involved.

Time horizon in economic theory

• Short run: the period of time in which the input of one or more productive agents is fixed
• Long run: the period of time in which all inputs are variable

actual length of the time horizon to use depends on

• the type of decision: when do the costs and benefits occur ?
• the expected life time of assets involved
• the time horizon for major transportation projects tends to be lengthy relative to that in other industries

The relationship between short and long run costs is explained by the envelope theorem. That is, the short run cost functions represent the behavior of costs when at least one factor input is fixed. If one were to develop cost functions for each level of the fixed factor the envelope or lower bound of these costs would form the long run cost function. Thus, the long run cost is constructed from information on the short run cost curves. The firm in its decision-making wishes to first minimize costs for a given output given its plant size and then minimize costs over plant sizes.

In the diagram below the relationship between average and marginal costs for four different firm sizes is illustrated. Note that this set of cost curves was generated from a non-homogeneous production function. You will note that the long run average cost function (LAC) is U-shaped thereby exhibiting all dimensions of scale economies.

Mathematically

${displaystyle Cleft(Qright)equiv C_{s}left(Q,Kleft(Qright)right)}$

${displaystyle {frac {partial Cleft(Qright)}{partial Q}}={frac {partial C_{s}left(Q,Kleft(Qright)right)}{partial Q}}+{frac {partial C_{s}left(Q,Kleft(Qright)right)}{partial K}}bullet {frac {partial Kleft(Qright)}{partial Q}}}$

where:

${displaystyle {frac {partial C_{s}left(Q,Kleft(Qright)right)}{partial K}}=0}$

provides the optimal plant size.

## Indicators of Aggregate Cost Behavior

Scale economies is the behavior of costs when the AMOUNT of an output increases while scope economies refers to the changes in costs when the NUMBER of outputs increases.

### Economies of Scale

Economies of scale refer to a long run average cost curve which slopes down as the size of the transport firm increases. The presence of economies of scale means that as the size of the transport firm gets larger, the average or unit cost gets smaller. Since most industries have variable returns to scale cost characteristics, whether or not a particular firm enjoys increasing, constant or decreasing returns to scale depends on the overall market size and the organization of the industry.

The presence or absence of scale economies is important for the industrial structure of the mode. If there were significant scale economies, it would imply fewer larger carriers would be more efficient and this, under competitive market circumstances, would naturally evolve over time. Scale economies are important for pricing purposes since the greater are the scale economies, the more do average and marginal costs deviate. It would, therefore, be impossible to avoid a deficit from long run marginal [social] cost pricing.

Another note of terminology should be mentioned. Economics of scale is a cost concept, returns to scale is a related idea but refers to production, and the quantity of inputs needed. If we double all inputs, and more than double outputs, we have increasing returns to scale. If we have less than twice the number of outputs, we have decreasing returns to scale. If we get exactly twice the output, then there are constant returns to scale. In this study, since we are referring to costs, we use economies of scale. The presence of economies of scale does not imply the presence of returns to scale.

Scale measures long-run (fully adjusted) relationship between average cost and output. Since a firm can change its size (network and capacity) in the long run, Economies of Scale (EoS) measures the relationship between average cost and firm size. EoS can be measured from an estimated aggregate cost function by computing the elasticity of total cost with respect to output and firm size (network size for the case of a transport firm).

#### Returns to Scale (Output Measure)

Increasing Returns to Scale (RtS)

${displaystyle f(tx_{1},tx_{2})>tf(x_{1},x_{2})}$

${displaystyle f(tx_{1},tx_{2})$

#### Economies of Scale (Cost Measure)

Economies of scale (EoS) represent the behavior of costs with a change in output when all factors are allowed to vary. Scale economies is clearly a long run concept. The production function equivalent is returns to scale. If cost increase less than proportionately with output, the cost function is said to exhibit economies of scale, if costs and output increase in the same proportion, there are said to be ‘constant returns to scale’ and if costs increase more than proportionately with output, there are diseconomies of scale.

• if cost elasticity < 1, or
${displaystyle LRMC$

-> increasing EoS

• if cost elasticity = 1, or
${displaystyle LRMC=LRAC}$

-> constant EoS

• if cost elasticity > 1, or
${displaystyle LRMC>LRAC}$

Economies of Density

There has been some confusion in the literature between economies of scale and economies of density. These two distinct concepts have been erroneously used interchangeably in a number of studies where the purpose was to determine whether or not a particular mode of transportation (the railway mode has been the subject of considerable attention) is characterized by increasing economies or diseconomies of scale. There is a distinction between density and scale economies. Density economies are said to exist when a one percent increase in all outputs, holding network size, production technology, and input prices constant, increase the firm’s cost by less than one percent. In contrast, scale economies exist when a one percent increase in output and size of network increases the cost by less than one percent, with production technology and input prices held constant.

Economies of density, although they have a different basis than scale economies, can also contribute to the shape of the modal industry structure. It can affect the way a carrier will organize the delivery of its service spatially. The presence of density economies can affect the introduction of efficient pricing in the short term, but generally not over the long term since at some point density economies will be exhausted. This, however, will depend upon the size of the market. In the air market, for example, deregulation has allowed carriers to respond to market forces and obtain the available density economies to varying degrees.

Returns to Density similar to returns to a capacity utilization when capacity is fixed in the short run. Since the plant size (network size for the case of transportation firms) is largely fixed in the short run, RTD measures the behavior of cost when increasing traffic level (output) given the plant size (network size). It is measured by the cost elasticity with respect to output.

• if cost elasticity < 1, or
${displaystyle SRMC$

-> increasing EoD

• if cost elasticity = 1, or
${displaystyle SRMC=SRAC}$

-> constant EoD

• if cost elasticity > 1, or
${displaystyle SRMC>SRAC}$

Economies of Capacity Utilization

A subtle distinction exists between economies of density, which is a spatial concept, and economies of capacity utilization, which may be aspatial. As a fixed capacity is used more intensively, the fixed cost can be spread over more units or output, and we have declining average cost, economies of scale. However, as the capacity is approached, costs may rise as delays occur. This gives a u-shaped cost curve.

While economies of scale refer to declining average costs, for whatever reason, when output increases; and economies of density refer to declining costs when output increases and the network mileage is held constant; economies of capacity utilization refers to declining costs as the percentage of capacity which is used increases, where capacity may be spatial or aspatial.

While density refers to how much space is occupied, capacity refers to how much a capacitated server (e.g. a bottleneck, the number of seats on a plane) is occupied, and may incorporate economies of density if the link is capacitated, such as a congesting roadway. However if a link has unlimited (or virtually unlimited) capacity, such as intercity passenger trains on a dedicated right-of-way at low levels of traffic, then economy of density is a more appropriate concept. Another way of viewing the difference is that economies of density refers to linear miles, while economies of utilization refer to lane miles.

### Economies of Scope

Typically, the transport firm produces a large number of conceptually distinct products from a common production facility. In addition, the products of most transportation carriers are differentiated by time, space and quality. Because a number of distinct non-homogeneous outputs are being produced from a common production facility, joint and common costs arise. The presence of joint and common costs give rise to economies of scope. There has been some confusion in the multi-product literature among the concepts of sub-additivity of the cost function, trans-ray convexity, inter-product complementarity and economies of scope. Sub additivity is the most general concept and refers to a cost function which exhibits the characteristic that it is less costly to produce different amounts of any number of goods in one plant or firm than to sub divide the products or service in any proportion among two or more plants. Trans-ray convexity is a somewhat narrower concept. It refers to a cost function which exhibits the characteristic that for any given set of output vectors, the costs of producing a weighted average of the given output vectors is no greater than the weighted average of producing them on a stand alone basis. Economies of scope refers to the cost characteristic that a single firm multi-product technology is less costly than a single product multi-firm technology. It, therefore, is addressing the issue of the cost of adding another product to the product line. Inter-product complementarity is a weak test of scope economies. It refers to the effect on the marginal cost of one product when the output of some other product changes. It, therefore, is changing the amount of output of two or more products and not the number of products. Whether scope economies exist and the extent to which they exist depend upon both the number of products and the level of each output. There have not been definitive empirical estimates of economies of scope for transportation modes which are based on reliable data and undertaken in a theoretically consistently fashion.

Thought Question: Most firms produce multiple products. Why do multiple product firms exist ?

It must be cheaper to have one firm to produce multiple products than have separate firms produce each type of product.

Economies of scope arise from shared or jointly utilised inputs, e.g., imperfectly divisible plant which if used to produce only one product would have excess capacity (freight and passenger services using same airplane, forward-back haul production using a truck or rail car, etc.).

This can be represented graphically as in the diagram on the right. In production space an isoquant would link two outputs and would have the interpretation of an isoinput line, that is, it would be the combination of outputs which are possible with a given amount of inputs. If there were economies of scope, the line would be concave to the origin, if there were economies of specialization it would be convex and if there were no scope economies it would be a straight line at 45 degrees.

Let

${displaystyle q=(q_{1},…,q_{n})}$

,

${displaystyle n}$

= number of different outputs.

Economies of scope exists if

${displaystyle c(q_{1})+…+c(q_{n})>c(q_{1},…,q_{n}).}$

${displaystyle q_{i}}$

, where

${displaystyle c(q)}$

is the cost for a firm to produce output

${displaystyle q}$

.

Scope economies are a weak form of ‘transray convexity’ and are said to exist if it is cheaper to produce two products in the same firm rather than have them produced by two different firms. Economies of scope are generally assessed by examining the cross-partial derivative between two outputs, how does the marginal cost of output one change when output two is added to the production process.

### Changes in Cost

Costs can change for any number of different reasons. It is important that one is able to identify the source of any cost increase or decreases over time and with changes in the amount and composition of output. The sources of cost fluctuations include:

• density and capacity utilization; movements along the short run cost function
• scale economies; movements along the long run cost function
• scope economies; shifts of the marginal cost function for one good with changes in product mix
• technical change which may alter the level and shape of the cost function

## Characterizing Transportation Costs

All modes of transport experience:

• economies of vehicle size up to a point
• increasing returns in provision of way and track capacity
• economies of longer distance travelled
• rapidly rising average cost with increased speed;
• exponentially increasing energy consumption with speed
• difficultly in identifying the costs associated with particular traffic because of indivisibilities in production and heterogeneity of output
• declining unit costs over a range of output because of indivisibilities,
• e.g. the backhaul problem, increase in traffic on the backhaul will reduce the average costs of the round trip operation
• indivisibilities in production give rise to “kinked” average cost curves and discontinuous marginal costs

## Costing

Costing is the method or process of ascertaining the relationship between costs and outputs in a way which is useful for making decisions (managerial, strategic, regulatory policy etc.). There are numerous examples where detailed cost information is necessary for carriers’ management decisions and government’s regulatory decisions. Also there are many carrier and government decisions requiring information about the behaviour of aggregate costs of a firm.

### Carrier Management Decisions

Requiring disaggregate cost info:

• rates and rate structure decisions;
• rate setting
• shipper-carrier negotiations
• financial viability of specific services; e.g.,
• rail passenger operations,
• rail branch lines
• decision to launch a specific service
• application of subsidies
• compensation for running rights;
• passenger trains
• leased right of way

Requiring aggregate cost info:

• carrier network plan
• plan for mergers and acquisitions
• strategic plan
• major investment decisions

### Policy Decisions

Requiring disaggregate cost info:

• enforcing pricing regulation
• decisions on public subsidies
• branchline abandonment decision – “short-line” sales
• user charges for government-owned infrastructure

Requiring aggregate cost info:

• decision on price and entry regulation;
• natural monopoly question – scale and density economies
• effect of regulation on efficiency;
• allocative efficiency
• X-efficiency
• approval of mergers and acquisition – scale and density economies
• decisions on transport infrastructure investment
• licensing of competitive services

## Aggregate Cost Analysis

Econometric cost functions are estimated to study the behaviour of aggregate costs in relation to the aggregate output level (economies of scale) and output mix (economies of scope).
The aggregate cost function also allows one to estimate the changes in productive efficiency over time.
This allows inference about the effect of regulation on productivity of an industry

### Which Costs

Economic theory suggests that costs are a function of at least factor prices and outputs. In practice, calculating costs, prices, or outputs can be tricky. For example, how should capital costs be determined ?

Capital costs may occur over one year but it is likely to be used over a long period of time. So we should use the opportunity costs which includes depreciation and interest costs. The capital stock of a firm will vary year to year.

Accountants tend to use historical costs which do not account for inflation. The point is that in the real world get all sorts of complications.

### Prices and Outputs

For prices and outputs, a firm may use many inputs and provide many different outputs. Transportation outputs are produced over a spatial network.
An appropriate definition of outputs is the movement of a commodity/passenger from an origin to a destination – a commodity/passenger trip. A trip from A to B is different from a trip from C to D (or B to A) even if the same distance. Ideally, a transport cost model should account for this multiproduct nature.
But cannot specify thousands of outputs -some aggregate is necessary. Often. lack of data requires aggregation to a single output measure like ton-kilometres or passenger-kilometres.

### Attribute Variables

To account for the multidimensional heterogeneous nature of outputs, one can use attribute variables such as average length of haul or average stage length. They will vary by firms. Operating characteristics such as average shipment size, average load factor, etc., also affect costs. For example, if plane or truck is not full, there is unused capacity; adding a commodity trip may incur little marginal cost; longer distances can lower AC by spreading terminal costs or takeoff fuel costs.

### Estimation

Cost function Estimation requires decisions on:

• short run vs. long run cost function
• short run cost functions from time series data;
• long run cost functions from cross-section data;
• variable vs. total cost function;
• variable cost functions are estimated by fixing some inputs such as physical plants (rail roadbed and track; aircraft fleet, etc)
• the choice of functional form
• the choice of output measure;
• single vs. multiple output measures
• revenue output vs. available output
• the choice of the level of aggregation of cost accounts
• the choice of attribute variables to account for heterogeneous nature of outputs being produced over time or across different firms in the sample data.

### Difficulties with Costing

• multi-dimensionality and heterogeneous nature of outputs
• indivisibilities in production
• costs may not occur at the same time as the outputs being produced. eg. capital costs may over one year but it is likely to be used over several years, and some expenses occur some time after the increases in outputs (expenses occur less frequently than changes in (train) trips), etc
• ambiguity in cost standards
• difficulty of relating past to future
• input price changes
• changes in production technology
• changes in operating conditions

## Disaggregate Costing

Disaggregate costing can be used to estimate the variable cost of a block of traffic, or traffic on a particular line,etc. it is useful for setting rates, investment decisions, subsidy determinations, etc. by companies themselves or government

Deductive (economic) vs inductive (engineering) approaches are used in transportation modeling, and analysis. The deductive approach uses modeling and prior relationships to specify a functional relationship which is then examined statistically. An inductive approach is based on a detailed understanding of physical processes.

Inductive Approaches

• Use of Engineering Relationships

Economic Approaches

• Average Cost Calculation using Accounting Info
• Statistical Costing

### Engineering Costing

Engineering costing focuses on the amount of each input required to produce a unit of output, or the technical coefficients of production. combining such coefficients with the costs of the inputs yields the cost function for the particular output.

There are two approaches to engineering costing:

• to derive the technical coefficients from physical laws or precise engineering relationships.
• to empirically establish the technical relationship by controlled experiment.
• shortcomings:
• data- and time-intensive costly
• nonstochastic
• must have well defined production processes

### Accounting Costing

• compiles the cost accounts categories relevant to the output or service in question, and use that information to estimate the costs associated with a specific movement.
• relatively cheap
• convenient
• shortcomings:
• data/information must exist
• the recorded values of assets may not be a reliable indicator of the actual opportunity costs of those assets
• the cost accounts may not distinguish fixed vs. variable costs Y over estimation of the marginal cost.
• the accounts are classified by the types of expenses, not by output type, it is difficult to uncover the true relations between cost and outputs
• the aggregation in the accounts may prevent identifying the costs which can be related to the production of particular outputs

### Statistical Costing

Statistical costing employes statistical techniques (usually multiple regression analysis) to infer cost-output relationships from a sample of actual operating experiences. It makes use of accounting information.

Basic steps in statistical costing are:

(1) Decompose and identify the intermediate work units associated with the specific traffic. For example, costing 500 tons of coal from point a to b, intermediate work units may consist of line haul, switching, terminal activities, administration, etc. Explanatory variables for these activities would include ton-miles, car-miles, yard-switching miles, train-hours, gallons of fuel, etc.

(2) Establish relationship between factor inputs and the intermediate process. This can be done by direct assignment of an expense category to the work unit, if causal relation is clear. Often, expenses are common to several types of traffic, so estimate statistical relationship with regression analysis.

For example, regression of track and roadway maintenance (TRM)

${displaystyle TRM=f({text{ton-miles, yard-switching minutes, train switching minutes, road miles}})}$

(3) Apply the marginal/unit costs of the intermediate work units estimated in step (2) to the work units identified in step (1).

(4) Sum all expenses in step (3) to calculate the total avoidable cost of a block of traffic.

## Evidence on Carrier Costs

How do the long run concepts of economies of scale and economies of scope and the short run concepts of economies of density and economies of capacity utilization influence costs? Why are they important to our discussion of transport infrastructure pricing? These questions will be addressed in the following section.

### Air Carriers

A considerable number of studies, Douglas and Miller (1974) [2], Keeler (1974) [3], Caves, Christensen and Tretheway (1984) [4], Caves, Christensen, Tretheway and Windle (1985)[5], McShan and Windle (1989)[6], and Gillen, Oum, and Tretheway (1985, 1990)[7][8], have been directed at determining the functional relationship between total per-unit operating costs and firm size in airlines. All studies have shown that economies to scale are roughly constant; thus, size does not generate lower per-unit costs. However, generally, the measures of economies of density illustrate that unit cost would decrease for all carriers if they carried more traffic within their given network. In other words, the industry experienced increasing returns to density. The results also indicated that the unexploited economies of density are larger for low density carriers.
Caves, Christensen, and Tretheway (1984) have shown that it is important when measuring costs to include a network size variable in the cost function, along with output, which would allow for the distinction between economies of scale and economies of density. McShan and Windle (1989) utilize the same data set as that used by Caves et al., and explicitly account for the hub and spoke configuration that has developed in the US since deregulation in 1978. They estimate a long run cost function which employs all the variables included in Caves et. al., and found economies to density of about 1.35. The hubbing variable indicates that, ceteris paribus, a carrier with 1% more of its traffic handled at hub airports expects to enjoy 0.11% lower cost than other similar carriers.

### Intercity Buses

Gillen and Oum (1984)[9] found that the hypothesis of no economies of scale can be rejected for the intercity bus industry in Canada; there are diseconomies of scale at the mean of the sample (0.91). Large firms were found to exhibit strong diseconomies of scale, and small and medium sized firms exhibit slight departures from constant returns. No cost complementarities are found to exist between the three outputs, namely, number of scheduled passengers, revenue vehicle miles of charter, tour and contract services, and real revenue from freight. These results, however, may be biased since no network measure was included in the estimating equations. The scale economy measure will, therefore, contain some of the influence of available density economies.

Since deregulation of the intercity bus industries in the US and the UK., the number of firms has been significantly reduced. In the absence of scale economies, the forces leading to this industry structure would include density economies. We have, for example, observed route reorganization to approximate hub-and-spoke systems and the use of smaller feeder buses on some rural routes.
The industry reorganization is similar to what occurred in the airline industry. The consolidation of firms was driven by density and not scale economies. One significant difference between these two industries, however, is airline demand has been growing while intercity bus demand is declining.

### Railway Services

The structure of railway costs is generally characterized by high fixed costs and low variable costs per unit of output. The essential production facilities in the railway industry exhibit a significant degree of indivisibility. As with other modes, the production of railway services give rise to economies of scope over some output ranges. For example, track and terminals used to produce freight services are also used to produce passenger services.

Caves, Christensen and Tretheway (1980)[10] have found that the US railway industry is characterized by no economies of scale over the relevant range of outputs. However, their sample does not include relatively small railroads, firms with less than 500 miles of track. Griliches (1972)[11] and Charney, Sidhu and Due (1977)[12] have found economies scale for such small US railroads. Friedlaender and Spady (1981) [13] suggested that there may be very small economies of scale with respect to firm size. Keeler (1974)[14], Harris (1977)[15], Friedlaender and Spady (1981) and Levin (1981)[16] have all shown that there are large economies of traffic density in the US railroad industry. They show that, allowing all factors of production except route mileage to vary, a railway producing 10 million revenue ton-miles per mile of road, for example, will have substantially lower average costs than will a railway producing only 5 million revenue ton-miles per mile of road. Harris (1977) estimated that approximately one-third of density economies were due to declining average capital costs, and two-thirds due to declining fixed operating costs, such as maintenance, and administration. Friedlaender and Spady (1981) estimate a short run cost function with five variable inputs, one quasi-fixed factor (structures) and two outputs which take the form of hedonic functions, accounting for factors such as low density route miles and traffic mixes. The study found no economies of scale. Caves, Christensen, Tretheway and Windle (1985) have examined economies of scale and density in the US railroads. Their basic result demonstrates that there are substantial economies of density in the US railway operations.

## Evidence on Infrastructure Costs

As early as 1962, Mohring and Harwitz[17] demonstrated that the financial viability of an infrastructure facility, under optimal pricing and investment, will depend largely upon the characteristics of its cost function. To quote Winston (1991)[18]: “ If capacity and durability costs are jointly characterized by constant returns to scale, then the facility’s revenue from marginal cost pricing will fully cover its capital and operating costs. If costs are characterized by increasing returns to scale, then marginal cost pricing will not cover costs; conversely, if costs are characterized by decreasing returns to scale, marginal cost pricing will provide excess revenue.”

The objective of this section is to provide a summary of the theoretical and empirical literature on the cost characteristics of modal infrastructure. The discussion will deal with the following types of infrastructure: airports, highways, and railways.

In developing a set of socially efficient prices for modes of intercity transport, it is not just the carrier’s cost structure which is important. Airports, roadways and harbors all represent public capital which is used by the carriers in the different modes to produce and deliver their modal services. This capital must also be priced in an efficient way to achieve the economic welfare gains available from economically efficient pricing. As with the carriers, the ability to apply first best pricing principles to infrastructure and still satisfy cost recovery constraints will depend upon the cost characteristics of building and maintaining the infrastructure.

As with carriers, the cost characteristics for infrastructure providers include scale economies, scope economies, density economies and utilization economies. Scale economies refer to the size of a facility; for example, is it cheaper to build three runways than it is to provide two runways? If so, there are economies of scale in the provision of runways. Scope economies encompass similar concepts as with carriers. Small, Winston and Evans (1989)[19] refer to scope economies in highways when both capacity and durability are supplied. Capacity refers to the number of lanes while durability refers to the ability to carry heavier vehicles. A similar concept would apply to airports: small and large aircraft, VFR and IFR traffic, and to harbors: large ships and small ships. Although rail infrastructure is currently supplied by the same firms operating the trains, there have been moves to separate infrastructure and carrier services. This separation will mean the track and terminals will have to be priced separately from carrier services.

Density economies should also, in principle, be evident in the provision of infrastructure. It is, for example, possible to expand outputs and all inputs for highways while holding the size of the network fixed.

Utilization economies refer to the short run cost function. They describe how quickly average and marginal costs will fall as capacity utilization approaches capacity. Although not of direct interest, they are important to consider in any cost estimation since failure to consider capacity utilization can bias upward the measures of both long run average and marginal costs.

### Airports

Economists have typically assumed that capacity expansion is divisible. Morrison (1983)[20], in his analysis of the optimal pricing and investment in airport runways, has shown that airport capacity construction is characterized by no economies of scale, and, therefore, under perfect divisibility of capacity expansion, the revenue from tolls will be exactly equal to the capital cost of capacity investment (Mohring and Harwitz, 1962). Morrison’s results, however, were based on a sample of 22 of the busiest airports in the US and did not include any small airports. In the literature, there is no empirical evidence on the cost characteristics of capacity construction of new small airports or capacity expansion of existing small airports (e.g. one runway).

### Highways

In general, highways produce two outputs: traffic volume which requires capacity in terms of the number of lanes, and standard axle loading which require durability in terms of the thickness of the pavement. Prior to determining economies of scale in this multi-product case, the measure of economies of scale for each output, or the product specific economies of scale, must be examined. Small, Winston, and Evans (1989) reported the existence of significant economies of scale associated with the durability output of roads, the ability to handle axle loads. This is because the pavement’s ability to sustain traffic increases proportionally more than its thickness. They also found evidence that there are slight economies of scale in the provision of road capacity; i.e. the capacity to handle traffic volume. However, they reported diseconomies of scope from the joint production of durability and capacity because as the road is made wider to accommodate more traffic, the cost of any additional thickness rises since all the lanes must be built to the same standard of thickness. They conclude that these three factors together result in highway production having approximately constant returns to scale. In other words, the output-specific scale economies are offset by the diseconomies of scope in producing them jointly.

### Railways

An important difference between rail and other modes of transportation is that most railroads provide the infrastructure themselves and the pricing is undertaken jointly for carrier services and infrastructure. However, in a few cases, ownership and/or management of the trackage has been separated from carriers. Sweden is a good example but even in the US there have been joint running rights on tracks. This creates a situation whereby one firm may be responsible for the provision of trackage and another for carrier services. It is, therefore, legitimate to ask if there are any scale economies in the provision of railway infrastructure. There are no empirical estimates but it may be possible to use some of the Small, Winston and Evans (1988) work for roads to shed some light on the issue.

Small et al. argue road infrastructure produces two outputs, durability and capacity. The former refers to the thickness of roads and the latter to their width. They found economies with respect to durability, but this is less likely to occur with a rail line since there would be a relatively broad range of rail car axle loading for a given level of durability of rail, ballast and ties. Thus, there may be some minor economies. The authors found diseconomies of scope from the joint production of durability and capacity for highways. These diseconomies are less likely to be evident in rail due to the broad range of durability noted above and the ability to restrict usage to specific tracks. On balance, it may be there are generally constant or minor economies in the provision of rail line infrastructure. The output specific scale economies seem to be minor as do the diseconomies of producing them jointly.

## Factors affecting Transportation Costs

Transportation costs seem to be rising, There are many factors which might explain this. These are listed below. This list is no doubt incomplete, but may serve as a point of discussion.

1. Standards
1. Standards have risen – Society now demands safety, features, environmental protection, access for the disabled, and quality that drive up the cost. Engineering design is often 20% of project costs. Does the firetruck really need to do a 360 degree turn on the cul-de-sac, or can it back out?
2. Smith’s Man of System – The man of system . . . is apt to be very wise in his own conceit; and is often so enamoured with the supposed beauty of his own ideal plan of government, that he cannot suffer the smallest deviation from any part of it. He goes on to establish it completely and in all its parts, without any regard either to the great interests, or to the strong prejudices which may oppose it. He seems to imagine that he can arrange the different members of a great society with as much ease as the hand arranges the different pieces upon a chess-board. He does not consider that the pieces upon the chess-board have no other principle of motion besides that which the hand impresses upon them; but that, in the great chess-board of human society, every single piece has a principle of motion of its own, altogether different from that which the legislature might chuse to impress upon it. If those two principles coincide and act in the same direction, the game of human society will go on easily and harmoniously, and is very likely to be happy and successful. If they are opposite or different, the game will go on miserably, and the society must be at all times in the highest degree of disorder. — Adam Smith, The Theory of Moral Sentiments, 1759
3. Gold-plating – Adding needless or useless features to projects. The costs of gold plating are several. Money spent on project X cannot be spent on project Y. This is the monetary opportunity cost of misallocation. Land devoted to project X cannot be devoted to project Y. More land also means greater distances to traverse. This is a spatial opportunity cost. There is a tension between the risk of gold plating (focus on benefits to the exclusion of cost) and of corner cutting (focusing on costs to the exclusion of benefits). But there is available to us a balance, building something which maximizes the difference between benefits and costs, not just looking at benefits or costs. Insufficient attention is placed on the trade-off, too much on the ends by advocates of one side or the other.
4. Design for forecast.
5. The State Aid system and associated standards – Funds are collected at the state and federal levels for transportation and then a portion of that money is transferred back to local governments for transportation. Along with the money comes requirements that dictate how that money is to be used. These include engineering requirements for things such as lane width, degree of road curvature and design speed and planning requirements for things like maintaining a hierarchical road network.
6. Doing construction on facilities still in operation. – Aside from the rare bridge, it is unnecessary to keep facilities opening and operating while doing construction. This reduces construction space, reducing time, increases set-up/break-down costs, and otherwise adds to total costs. Construction is much faster (and thus cheaper) if rebuilding could be done on a closed facility. See the w: Tube Lines as the classic example of the high cost of doing construction only at night and weekends, but keeping the line in operation. The system as a whole must be reliable, meaning I can get from here to there, but that does not mean every segment must be open 24/7/365. One reason the reconstruction of the I-35W bridge was so fast as that they contractors did not need to worry about existing traffic, (and it was design/build).
7. Environmental Impact Statements (Reports) lead to “lock-in”
8. Open government/costs of democracy – The planning process is required by law to bring in as many stakeholders as possible. This has (potentially) led to transportation investment being sought and justified for non-transportation concerns. Transportation investment is now used for social, moral and economic goals that are not directly related to mobility.
9. Climate change adaptation is increasing the costs of projects.
2. Scale economies
1. There are in-sufficient economies of scale – When everything is bespoke, there is no opportunity for standardization and economies of scale. While many rail against cookie-cutter design, it is only with cookie-cutters that we get lots of cookies.
2. Thin markets – There is no online department store for public works. I cannot go online and buy a transit bus or an interchange. The internet has not driven down prices in this field the way it has in so many others. As a result a few vendors can collude or orchestrate higher prices than would be faced in a more competitive market.
3. Peaking – Transportation agencies attempt to provide high levels of peak capacity to accommodate the demand that results from un-priced roads and highways. This is very costly capacity to provide. If tolls were charged that reflected true costs people would drive less, especially during peak hours. It would therefore cost much less to provide the economically optimal amount of peak system capacity.
3. Change of scope
1. Projects are scoped wrong – We have investments that don’t match actual demands. And this is not just for megaprojects. We have big buses serving few passengers. We have overgrown highways. We have a fear of building too small and having congestion or crowding so we build too big.
2. Project creep – Side-payments in project development: noise walls hither and thither, etc. Side-payments are a required part of the politics of getting something built.
3. “Starchitecture”,
4. Fragmented governance leads to large and meandering projects rather than centralized projects. Politicians have to “share the wealth” of projects. This is perhaps a cause of “project creep.”
4. Principal-agent problems
1. Other people’s money – Public works agencies are spending Other People’s Money, and so are less motivated to get value for dollar than an individual consumer on their own. This principal-agent problem prevails in lots of organizations, but especially so in public works where the bias is not to have a failure. There was an old saying in business, no one ever got fired for buying IBM. The same holds in public works, where rocking the boat with new or innovative technologies is not sufficiently rewarded.
2. Benefits are concentrated, costs are diffuse – As a result, the known beneficiaries lobby hard for projects, but not just to build it, but to build it in a way that is expensive. Costs are diffuse, it is seldom worth the taxpayer’s time to oppose a project just because of its costs, which are spread among millions of other taxpayers. See: w:The Logic of Collective Action.
3. Decision-makers are remote – Remote actors cannot have precise information about local conditions, and in the absence of a free market in transportation (there is generally one buyer, who is generally a government agency), prices are not clear. As a result these remote actors misallocate because they are misinformed. This notion derives from the w: Economic calculation problem See w:The Fatal Conceit.
4. Benefit cost analysis is only as good as the integrity of the data and the analysts.
5. B/C analysis is not used to affect project outcomes.
6. Planners and engineers are paid as percentage of total project cost.
7. Formula spending reduces the incentive or need to worry much about costs. This is obviously related to many of the other hypotheses already considered but I think deserves it’s own number.
8. Lack of user fee funding – projects funded out of user fees are more likely to be efficient, partly because the agencies or private parties receiving those fees know the fees are limited and partly because they want to spend them in ways that will generate more fees (which means in ways that benefit users enough that the users are willing to pay for them).
9. Federal funds favor capital-heavy technologies and investments. – Federal funding programs create perverse incentives that lead to very costly capital projects. Almost any project looks good if somebody else is paying for most of it. For example every year billions of dollars are spent on passenger rail projects that would never be funded were it not for generous Federal grants. Just look at the high speed rail program or the FTA New Starts program. There are examples on the highway side too, such as bridges to nowhere and freeways in rural areas with little traffic. These Federal programs, no matter how well intentioned, tip the local decision making process in favor of expensive capital projects and discourage consideration of lower cost options and policy reforms.
10. Public ownership – Most of the transportation system is owned, planned, and managed by public agencies. These entities have many objectives but efficiency and cost-effectiveness are rarely a high priority. The public sector does some things well but it doesn’t usually do them very efficiently. As a result transportation revenues are not always efficiently converted to transportation user benefits.
11. Multi-jurisdiction – Because transportation involves a large number of public agencies with overlapping or intertwined responsibilities planning is complex and inefficient. Projects end up with all the bells and whistles needed to satisfy the agencies and constituencies that could block a proposal. Local elected officials often load up regional plans with pet projects that do little to improve transportation system performance. There is a whole science to how public agencies bargain with each other and interact, unfortunately the results are rarely optimal from a cost-effectiveness perspective. The principal/agent problem is part of the reason for this, but only a part. In nearly every metropolitan area in the United States institutional structure results in transportation plans and policies that fall far short of the cost-effectiveness that could be achieved.
12. Graft.
13. Poor commissioning – Contracts determining who does what on a project are poorly written, and affect outsourced projects.
14. Separation of design and build – Different firms are responsible for engineering and construction, creating high communication costs.
15. Union work rules (not wages) that inhibit productivity gains through new technologies.
5. Project Duration
1. Paralysis by analysis – The bureaucratic requirement to do analysis delays projects and adds costs
2. Lack of upfront funds – Delays projects adds to ultimate costs.
3. Lack of consensus – Political requirements for consensus add delays.
4. Mismanagement and graft add to delays.
6. Other
1. The highest demand areas for maintenance and new stock occur in places that are expensive.
2. Envy – is a much bigger problem in public works than in personal life. – I pay taxes for those things, why does Jurisdiction X get an LRT when my neighborhood/district doesn’t? It’s a recipe for political hostages at budget time, as few political leaders have any reason to say “You know, the benefit cost on a project in my district just shows the project makes no sense.” It’s leads to two problems: projects that make no sense to serve some notion of geo-political equity, and project creep because if Jurisdiction X’s light rail stations had public art and golden knobs and a fountain, then my district’s light rail should have those and more. Combined with the Other People’s Money problem, this type of envy is a recipe for project creep.
3. Materials are scarcer (and thus more expensive).
4. Stop/start investment.
5. w:Ratchet effect – Interest groups are attracted to a particular public issue and pressure the legislative body to increase spending on that issue, but make it impossible to decrease spending on the issue.
6. w:Baumol’s cost disease – The rise of wages in jobs without productivity gains is caused by the necessity to compete for employees with jobs that did experience gains and hence can naturally pay higher salaries.
7. Transit investment isn’t realizing any productivity gains from labor. – Every dollar spent on public transportation yields 70% more jobs than a dollar spent on highways. This is used to bolster the argument that we should spend more on transit, but instead suggests we are much better at building roads than at building transit. As labor is a large proportion of total cost, transit investment has not realized productivity gains that have occurred in road building. This could be explained in part by lack of competition, low levels of total investment haven’t brought new producers into the market, or a number of other reasons. I don’t think the relatively high number of jobs per dollar spent necessarily means that transit investment is more virtuous. It may just be more inefficient. This is a problem with treating transport investment as industrial policy.
8. Utility works are uncharged.
9. Experience and Competence – The US has no experience with high-speed rail, so there is no domestic expertise.
10. Ethos, training and prestige – Transportation engineering is more prestigious in other countries.
11. Government power – Governments have more power to implement in other countries.
12. Legal system – Legal systems are more amenable to infrastructure construction, including liability, bonds, and insurance.

## References

1. Spulber, D. (1989), Regulation and Markets, The MIT Press Cambridge, Massachusetts.
2. Douglas, G. and J. Miller (1974), Economic Regulation of Domestic Air Transport: Theory and Policy, Brookings Institution, Washington, D.C.
3. Keeler, T. (1974), “Railroad Costs: Returns to Scale and Excess Capacity”, Review of Economics and Statistics, 56, 201-208.
4. Caves, D., L. R. Christensen, and M. W. Tretheway (1984), “Economies of Density versus Economies of Scale: Why Trunk and Local Service Airline Costs Differ”, Rand Journal of Economics, 15 (4), Winter, 471-489.
5. Caves, D., L. R. Christensen, M. W. Tretheway, and R. Windle (1985), “Network Effects and the Measurement of Returns to Scale and Density for U.S. Railroads”, in Analytical Studies in Transport Economics, Daughety (ed).
6. McShan, S. and R. Windle (1989), “The Implications of Hub-and-Spoke Routing for Airline Costs and Competitiveness”, Logistics and Transportation Review, 35 (3), September, 209-230.
7. Gillen, D.W., T.H. Oum, and M.W. Tretheway (1985), Airline Cost and Performance: Implications for Public and Industry Policies, Centre for Transportation Studies, University of British Columbia, Vancouver, B.C., Canada.
8. Gillen, D.W., T.H. Oum, and M.W. Tretheway (1990), “Airline Cost Structure and Policy Implications: A Multi-Product Approach for Canadian Airlines”, Journal of Transport Economics and Policy, Jan, 9-33.
9. Gillen, D.W. and T.H. Oum (1984), “A Study of Cost Structures of The Canadian Intercity Motor Coach Industry”, Canadian Journal of Economics, 17(2), May, 369-385.
10. Caves, D., L. R. Christensen, and M. W. Tretheway (1980), “Flexible Cost Functions for Multi-product Firm”, Review of Economics and Statistics, August, 477-481.
11. Griliches, Z. (1972), “Cost Allocation in Railroad Regulation”, Bell Journal of Economics and Management Science, 3, 26-41.
12. Charney, A., N. Sidhu, and J.Due (1977), “Short Run Cost Functions for Class II Railroads”, Logistics and Transportation Review, 17, 345-359.
13. Friedlaender, A. F. and R. H. Spady (1981), Freight Transport Regulation: Equity, Efficiency and Competition in the Rail and Trucking Industries, MIT Press, Cambridge, Mass.
14. Keeler, T. (1974), “Railroad Costs: Returns to Scale and Excess Capacity”, Review of Economics and Statistics, 56, 201-208.
15. Harris, R. (1977), “Economics of Traffic Density in the Rail Freight Industry”, Bell Journal of Economics, 8, 556-564.
16. Levin, R. (1981), “Railroad Rates: Profitability and Welfare Under Regulation”, Bell Journal of Economics, 11, 1-26.
17. Mohring, Herbert D. and I. Harwitz (1962), Highway Benefits: An Analytical Framework, Northwestern University Press, Evanston, Illinois.
18. Winston, C. (1991), “Efficient Transportation Infrastructure Policy”, Journal of Economic Perspectives, 5 (1), Winter, 113-127.
19. Small, K., C. Winston, and C. Evans (1989), Road Work: A New Highway Pricing and Investment Policy, Brookings Institute, Washington, D.C.
20. Morrison, S. (1983), “Estimating the Long Run Prices and Investment Levels for Airport Runways”, in T. Keeler (ed), Research in Transportation Economics. Vol.1, 103-131.

An externality is a cost or benefit incurred by a party’s decision or purchase on another, who neither consents, nor is considered in the decision. One example of a negative externality we will consider is pollution

## Introduction

There has been a long-standing interest in the issue of the social or external costs of transportation (see for instance: Keeler et al. 1975 [1], Fuller et al. 1983 [2], Mackenzie et al. 1992[3], INRETS 1993 [4], Miller and Moffet 1993 [5], IWW/INFRAS 1995 [6], IBI 1995 [7]). The passions surrounding social costs and transportation, in particular those related to the environment, have evoked far more shadow than light. At the center of this debate is the question of whether various modes of transportation are implicitly subsidized because they generate externalities, and to what extent this biases investment and usage decisions. On the one hand, exaggerations of environmental damages as well as environmental standards formulated without consideration of costs and benefits are used to stop new infrastructure. On the other hand, the real social costs are typically ignored in financing projects or charging for their use.

Associated with the interest in social and external cost has been a continual definition and re-definition of externalities in transportation systems. Verhoef (1994)[8] states “An external effect exists when an actor’s (the receptor’s) utility (or profit) function contains a real variable whose actual value depends on the behavior of another actor (the supplier) who does not take these effects of his behavior into account in this decision making process.” This definition eliminates pecuniary externalities (for instance, an increase in consumer surplus), and does not include criminal activities or altruism as producers of external benefits or costs. Rothengatter (1994) [9] presents a similar definition: “an externality is a relevant cost or benefit that individuals fail to consider when making rational decisions.” Verhoef (1994) divides external cost into social, ecological, and intra-sectoral categories, which are caused by vehicles (in-motion or non-in-motion) and infrastructure. To the externalities we consider (noise, congestion, crashes, pollution), he adds the use of space (e.g. parking) and the use of matter and energy (e.g. the production and disposal of vehicles and facilities). Button (1994) [10] classes externalities spatially, considering them to be local (noise, lead, pollution), transboundary (acid rain, oil spills), and global (greenhouse gases, ozone depletion). Gwilliam and Geerlings (1994) [11] combines Verhoef’s and Button’s schemes, looking at a Global, Local, Quality of Life (Social), and Resource Utilization (air, land, water, space, materials) classification.

Rothengatter (1994) views externalities as occurring at three levels: individual, partial market, total market, and argues that only the total market level is relevant for checking the need of public interventions. This excludes pecuniary effects (consumer and producer surplus), activities concerning risk management, activities concerning transaction costs. Externalities are thus public goods and effects that cannot be internalized by private arrangements.

Rietveld (1994) [12] identifies temporary effects and non-temporary effects occurring at the demand side and supply side. Maggi (1994) divides the world by mode (road and rail) and medium (air, water, land) and considers noise, crashes, and community and ecosystem severance. Though not mentioned among the effects above, to all of this might be added the heat output of transportation. This leads to the “urban heat island” effect — with its own inestimable damage rate and difficulty of prevention.

Coase (1992) [13] argues that the problem is that of actions of firms (and individuals) which have harmful effects on others. His theorem is restated from Stigler (1966) [14] as “… under perfect competition, private and social costs will be equal.” This analysis extends and controverts the argument of Pigou (1920) [15], who argued that the creator of the externality should pay a tax or be liable. Coase suggests the problem is lack of property rights, and notes that the externality is caused by both parties, the polluter and the receiver of pollution. In this reciprocal relationship, there would be no noise pollution externality if no-one was around to hear. This theory echoes the Zen question “If a tree falls in the woods and no-one is around to hear, does it make a sound?”. Moreover, the allocation of property rights to either the polluter or pollutee results in a socially optimal level of production, because in theory the individuals or firms could merge and the external cost would become internal. However, this analysis assumes zero transaction costs. If the transaction costs exceed the gains from a rearrangement of activities to maximize production value, then the switch in behavior won’t be made.

There are several means for internalizing these external costs. Pigou identifies the imposition of taxes and transfers, Coase suggests assigning property rights, while our government most frequently uses regulation. To some extent all have been tried in various places and times. In dealing with air pollution, transferable pollution rights have been created for some pollutants. Fuel taxes are used in some countries to deter the amount of travel, with an added rationale being compensation for the air pollution created by cars. The US government establishes pollution and noise standards for vehicles, and requires noise walls be installed along highways in some areas.
Therefore, a consensus definition might be, “Externalities are costs or benefits generated by a system (in this case transportation, including infrastructure and vehicle/carrier operations,) and borne in part or in whole by parties outside the system.”

## Definitions

An externality is that situation in which the actions of one agent imposes a benefit or cost on another economic agent who is not party to a transaction.

Externalities are the difference between what parties to a transaction pay and what society pays

• A pecuniary externality, increases the price of a resource and therefore involves only transfers,
• A technical externality exhibits a real resource effect. A technical externality can be an external benefit (positive) or an external disbenefit (negative).

### Examples

Negative externalities (external disbenefits) are air pollution, water pollution, noise, congestion.

Positive externalities (external benefits) include examples such as bees from apiary pollinating fruit trees and orchards supplying bees with nectar for honey.

The source of externalities is the poorly defined property rights for an asset which is scarce. For example, no one owns the environment and yet everyone does. Since no one has property rights to it, no one will use it efficiently and price it. Without prices people treat it as a free good and do not cost it in their decision making. Overfishing can be explained in the same way.

We want that amount of the externality which is only worth what it costs. Efficiency requires that we set the price of any asset >0 so the externality is internalized. If the price is set equal to the marginal social damages, we will get a socially efficient amount of the good or bad. Economic agents will voluntarily abate if the price is non-zero.

The Coase Theorem states that in the absence of transaction costs, all allocations of property are equally efficient, because interested parties will bargain privately to correct any externality. As a corollary, the theorem also implies that in the presence of transaction costs, government may minimize inefficiency by allocating property initially to the party assigning it the greatest utility.

## Pareto Optimality

A change that can make at least one individual better off, without making any other individual worse off is called a Pareto improvement: an allocation of resources is Pareto efficient when no further Pareto improvements can be made.

## Thought Question

What is the Optimal Amount of Externality?

(Is it zero? Why or why not?)

## Damages vs. Protection

Social Costs: Damages vs. Protection

Social Costs: Supply and Demand

The tradeoff between benefits and costs is central to most economic analyses. Costs and benefits are both measurable and immeasurable, and a complete analysis must consider transaction and information costs as well as market costs. Individuals strive to maximize net benefits (benefits after considering costs), society might apply this to social costs as well. Reducing damages requires increasing protection (defense, abatement, or mitigation) to attenuate the damage. At some point, the cost of protection outweighs the benefit of reducing residual damages. This is illustrated in the figure. Whether this point is at zero damages (no damage is acceptable), zero protection (the damage is so insignificant as to be irrelevant), or somewhere in between is an empirical question. Total social costs are minimized where the marginal cost of additional damages equals the cost of additional protection. Whether the marginal costs of damage and of protection are fixed, rising or declining with output, and by how much will be another important empirical question.

The notion of damages and protection is compatible with the idea of supply and demand, as illustrated in the figure to the right. Here, the change in damages with output (

${displaystyle dD/dQ}$

) is the demand curve (the marginal willingness to pay to avoid damage), and the change in protection (attenuation) with output is the supply curve (marginal cost) and represented as (

${displaystyle dA/dQ}$

). Again, the slopes of the curves are speculative:

In the next Figure, area A represents the consumer surplus, or the benefit which the community receives from production, and is maximized by producing at qo (marginal cost of protection or attenuation equals the marginal cost of defense). The shaded area B represents production costs, and is the amount of social cost at the optimal level of production. Area C is non-satisfied demand, and does not result in any social costs so long as production remains at

${displaystyle q_{o}}$

.

## Systems

Social Costs: Causes and Effects. Source:[16]

Central to the definition and valuation of exernalities is the definition of the system in question. The intercity transportation system is open, dynamic, and constantly changing. Some of the more permanent elements include airports, intercity highways , and railroad tracks within the state. The system also includes the vehicles using those tracks (roads, rails, or airways) at any given time. Other components are less clear cut – are the roads which access the airports, freeways, or train stations part of the system? The energy to propel vehicles is part of the system, but is the extraction of resources from the ground (e.g. oil wells) part of the system? DeLuchi (1991) [17] analyzes them as part of his life-cycle analysis, but should we? Where in the energy production cycle does it enter the transportation system?

Any open system influences the world in many ways. Some influences are direct, some are indirect. The transportation system is no exception. Three examples may illustrate the point:

1. Cars on roads create noise—this we consider a direct effect.
2. Roads reduce the travel time between two places, which increases the amount of land development along the corridor—this is a less direct effect, not as immediate or obvious as the first. Other factors may intervene to cause or prevent this consequence.
3. The new land development along the corridor results in increased demand for public schools and libraries—this is clearly an indirect effect of transportation.

As can be seen almost immediately, there is no end to the number or extent of indirect effects. While recognizing that the economy is dynamic and interlinked in an enormous number of ways, we also recognize that it is almost impossible to quantify anything other than proximate, first order, direct effects of the transportation system. If the degree to which “cause” (transportation) and “effect” (negative externality) are correlated is sufficiently high, then we consider the effect direct; the lower the probability of effect following from cause, the less direct is the effect. The question of degree of correlation is fundamentally empirical.

On the other hand, this raises some problems. Automobiles burn fuel that causes pollution directly. Electric powered high speed rail uses energy from fuel burned in a remote power plant. If the electricity is fully priced, including social costs, then there is no problem in excluding the power plant. But, if the social costs of burning fuel in a power plant are not properly priced, then to ignore these costs would be biased. This is the problem of the first best and second best. The idea of the first best solution suggests that we optimize the system under question as if all other sectors were optimal. The second best solution recognizes that other systems are also suboptimal. Clearly, other systems are suboptimal to some extent or another. However, if we make our system suboptimal in response, we lessen the pressure to change the other systems. In so doing we effectively condemn all other solutions to being second best.

Button (1994) develops a model relating ultimate economic causes to negative externalities and their consequences as summarized in the following graphic. Users and suppliers do not take full account of environmental impacts, leading to excessive use of transport. Button argues that policy tools are best aimed at economic causes, but in reality measures are aimed at any of four stages. Here we are considering the middle stage, physical causes and symptoms, and are ignoring feedback effects.

Another view has the “externalities” as inputs to the production of transportation, along with typical inputs as construction of transportation and the operation and maintenance of the system. There are multiple outputs, simplified to person trips and freight trips, although of course each person trip is in some respects a different commodity. This view comports with Becker’s (1965) [18] view that households use time in the production of commodities — of which travel might be one.

To establish optimal emission levels for pollution, congestion or any other externality consider the following framework. All emissions, damages, and costs are from one or more sources

${displaystyle i}$

, subscripts are omitted for clarity.

${displaystyle C(a)=C(z-e),!}$

where:

• ${displaystyle C(a),!}$

is the Cost of Abatement function with

• ${displaystyle C(a)’>0,!}$

${displaystyle C(a)”geq 0,!}$

and

${displaystyle D=D(e),!}$

where:

• ${displaystyle D(e),!}$

is the Cost of Damages function due to emissions .

• ${displaystyle e,!}$

is the actual emission from source.

• ${displaystyle z,!}$

is the amount of emission at source in an uncontrolled state.

• ${displaystyle a=z-e,!}$

is the amount of abatement at source.

Note if

${displaystyle a=0,!}$

, and thus

${displaystyle z=e,!}$

the actual emissions equal the maximum amount possible.

The solution to the problem is to minimize the sum of damage costs and abatement costs or

${displaystyle min D(e)+C(z-e),!}$

${displaystyle {frac {delta D}{delta e}}={frac {delta C}{delta a}},!}$

which indicates a constant marginal demand function.

The following is true:

${displaystyle {frac {delta D}{delta e}}={frac {delta C}{delta a}}bullet {frac {partial a}{partial e}},!}$

${displaystyle therefore {frac {delta D}{delta e}}={frac {delta C}{delta a}},!}$

This states that the optimal amount of any externality is established by minimizing the sum of damage and abatement costs so we end up with E* amount of aggregate pollution.

If a profit maximizing firm were faced with an abatement charge they would internalize the externality or abate until the marginal cost of abatement were equal to the price of pollution or the change.

## Government Standards

If the government wanted to establish a ‘standard’ it would require knowledge of:

• level of marginal damages
• Marginal Cost function of polluters

It would therefore appear that there is an informational advantage to pricing.

The solution which has been illustrated above also applies with:

• spatially differentiated damages
• non-linear damage functions
• non-competitive market settings

Standards dominate charged because

• Uncertainty with respect to the marginal damage function.
• Uncertainty with respect to the marginal abatement costs.

### Uncertainty with respect to marginal damage function

Now consider the situation where the MC of abatement has been underestimated so the true MC of abatement lies above the estimated MC of abatement function. Consider a standards scheme. Using the estimated MC of abatement the emission level is set at e instead of e*. Thus, the emission level is too low relative to the optimum. With the level of abatement too high, the damages reduced due to having this lower level of emissions is eAce* but at the cost of much higher abatement costs of eBCe*. The net social loss will be ABC.

Alternatively, suppose the authority set a sub-optimal emission standard of e because it is using the erroneous MD function. With emissions at e rather than e*, we again end up with a net social loss of ABC. Therefore, uncertainty with respect to the marginal damage function provides NO ADVANTAGE to either scheme; pricing or standards.

### Uncertainty with respect to marginal abatement costs

Now consider the situation where the MC of abatement has been underestimated so the true MC of abatement lies above the estimated MC of abatement function. Consider a standards scheme. Using the estimated MC of abatement the emission level is set at e instead of e*. Thus, the emission level is too low relative to the optimum. With the level of abatement too high, the damages reduced due to having this lower level of emissions is eAce* but at the cost of much higher abatement costs of eBCe*. The net social loss will be ABC

Now consider a pricing scheme. The authority would set the emission charge at EC by setting the MD function equal to the MC of abatement function. This would result in a level of emission of e; thinking this is the correct amount. But with a true MC of abatement at MCT the level of emissions which the charge EC will generate will be e’.

e’ > e* so we have too high a level of emissions. Pollution damages will increase by the amount e*CDe’ but the abatement costs will be reduced (because of higher allowed emissions) by e*CEe’. Therefore, the net social loss will be CDE.
Generally, there is no reason to expect CDE = ABC but it has been shown that

${displaystyle WL_{T}=WL_{q}=0.5left({frac {EC}{E}}right)bullet left(Delta eright)^{2}left({frac {1}{varepsilon _{D}}}+{frac {1}{varepsilon _{C}}}right)}$

where:

The welfare loss from pricing and standards will be equal if
1. = in absolute value
or
2.

${displaystyle Delta e=0}$

or

${displaystyle MC_{A}=MC_{T}}$

Standards will be preferred to charges when

${displaystyle WL_{T}-WL_{q}>0}$

${displaystyle left|varepsilon _{D}right|$

.

If

${displaystyle varepsilon _{C}>0}$

${displaystyle varepsilon _{D}>0}$

Rationale

The rationale for this is:

• if the MD function is steep (e.g. with very toxic pollution) even a slight error in
${displaystyle e}$

will generate large damages. With uncertainty about costs, the chances of such errors is greater with a charging scheme.

• if the MD function is flat, a charge will better approximate marginal damages. If the damage function is linear, the optimal result is independent of any knowledge about costs.
• if the MC is steep, an ambitious standard could result in excessive costs to abators. A charge places an upper limit on costs.

Therefore, the KEY in this is charges set an upper limit on costs while standards set an upper limit on discharges.

Externality prices can take three forms:

1. use to optimize social surplus

2. use to achieve a predetermined standard at least cost]

3. use to induce compliance to a particular standard

Perhaps the best know ‘cure’ for the congestion externality facing most major cities has been advocated by economists; road pricing. Standards are achieved in this instance by continuing to build roads.

## Measurement

The cost of an externality is a function of two equations. The first relates the physical production of the externality to the amount of transportation output. The second computes the economic cost per unit of externality. The amount of an externality produced by transportation is the result of the technology of the transportation, as well as the amount of defense and abatement measures undertaken. There are several issues of general concern in the physical production of externalities. They are classified as: fungibility, geography, life cycle, technology, and point of view. Each are addressed in turn.

### Fungibility

“Is the externality fungible?” In other words, does the externality which is physically produced by the system under question have to be eliminated or paid for, or can something substitute for it. For example, a car may produce X amount of Carbon Dioxide. If carbon dioxide were not fungible, then that X would need to be eliminated, or a tax assessed based on the damage that X causes. However, if it were fungible, then an equivalent amount X could be eliminated through some other means (for instance, by installing pollution control on a factory or by planting trees). The second option may be cheaper, and this may influence the economic effects of the pollution generated.

### Geography

“Over what area are the externalities considered?” “Is a cost generated by a project in California which is borne by those outside California relevant?” This is particularly important in estimating environmental costs, many of which are global in nature. If we try to estimate damages (rather than the protection costs of defense, abatement, and mitigation), this becomes particularly slippery. However, if we can assume fungibility, and use the cost of mitigation techniques, the measurement problem becomes much simpler. Ideally, we would obtain estimates for both protection and damages in order to determine the tradeoffs.

### Life Cycle

In some respects we would like to view the life-cycle of the transportation system. But it becomes more difficult to consider the life-cycle of every input to the transportation system. The stages which may be considered include: Pre-production, construction, utilization, refurbishing, destruction, and disposal. Ignoring the life-cycle of all inputs may create some difficulties. Electric power will produce pollution externalities at production in a power plant, before it enters the transportation system. Thus, modes using electric power (rail, electric cars), would be at an advantage using this decision rule over modes which burn fuel during the transport process (airplanes, gasoline powered cars, diesel trains). This is true, though to a lesser extent, with other inputs as well.

### Technology

The technology involved in transportation is constantly changing. The automobile fleet on the ground in 2000 will have very different characteristic than that in the year 1900 regarding the number of externalities produced. Hopefully, cars will be safer, cleaner, and quieter. Similar progress will no doubt be made in aircraft and trains. While the analysis will initially assume current technology, sensitivity tests should consider the effect that an improved fleet will have on minimizing externality production.

### Macro vs. Micro Analysis Scale

Estimates for externalities typically come in two forms macro and micro levels of analysis. Macroscopic analysis uses national (or global) estimates of costs as share of gross domestic product (GDP), such as Kanafani (1983), Quinet (1990), and Button (1994). The data for microscopic analysis is far more dispersed. It relies on numerous engineering and empirical cost-benefit and micro-economic studies. By and large, this study is a microscopic analysis, though, on occasion, the macroscopic numbers will be used as benchmarks for comparison and estimates of data where not otherwise available. This will be true for both the physical production of externalities as well as their economic costs through damages borne or protection/attenuation measures.
Once cost estimates are produced, they can be expanded to estimate the state-wide social costs of transport as a share of state product (California GDP), which can be compared with other national estimates.

Two important issues of concern in measuring the economic cost of externalities are: the basis over which the output is measured and the consistency of the measurement . When estimating the full cost of externalities, the amount of externality is not simply the amount of traffic on the road multiplied by some externality rate. Rather, it must be measured as the difference between what is generated systemwide with and without the facility. For instance, a new freeway lane will have several effects: diverting existing traffic from current facilities, inducing new traffic on the new facility, and inducing new/different traffic on the old facility. The amount of this change must be accurately determined with a general equilibrium approach to estimate demand. In a general equilibrium approach, the travel time/cost used to estimate the amount of demand is equal to the travel time/cost resulting from that demand. Switching traffic from an older facility to a newer facility may in fact reduce the amount of negative externalities generated. For instance, the number of accidents or their severity may decline if the new facility is safer than the old. On the other hand, the induced traffic, while certainly a benefit in that it increases commerce, also imposes new additional costs, more accidents, pollution and noise. It is the net change which must be considered.

When addressing the costs of externalities, the estimates used across all externalities should be consistent. Cost estimates contain implicit assumptions, particularly concerning the value of time, life, and safety. Key questions can be asked of any study:

• Is the value of life and health used in estimating the cost of accidents the same as used in estimating the human effects of pollution?
• Is the value of time used consistent between congestion costs and accidents? With congestion, many are delayed a small time, crashes (ignoring congestion implications), a few are delayed a long time.

## Cost-Function Estimation Methods

Many approaches have been undertaken to estimate the costs of externalities. The first class of approaches we call “Damage” based methods, the second can be called “Protection” based methods. The damage based methods begin with the presumption that there is an externality and it causes X amount of damage through lower property values, quality of life, and health levels.

The protection methods estimate the cost to protect against a certain amount of the externality through abatement, defense, or mitigation. One example of a defense measure is thicker windows in a house to reduce noise from the road. An abatement measure would have the highway authority construct noise walls to reduce noise or require better mufflers on vehicles. A mitigation measure may only be applicable for certain types of externalities; e.g. increased safety measures that reduce accidents on one facility also offset the increased number of accidents on another facility.

Rising marginal costs are expected of protection measures. The first quantity of externality abated /defended/mitigated is cheaper than the second and so on because the most cost-effective measures are undertaken first. This is not to say there are no economies of scale in mitigating externalities within a given mitigation technology. It merely suggests that between technologies, costs will probably rise.
The mitigation approach can be applied if we consider the externality fungible. Air pollution from the road may cause as much damage as an equivalent amount of pollution from nearby factories. The most cost effective approach to eliminating the amount of pollution produced by the road may come from additional scrubbers on the factory. While it may be prohibitively expensive to eliminate 100% of roadway pollution from the roadway alone, it may be quite reasonable to eliminate the same amount of pollution from the system. Determining the most effective method of mitigating each system-wide externality requires understanding the nature of its fungibility.

Neither of these two approaches (Damages or Protection) will necessarily produce a single value for the cost of a facility. It is more likely that each approach will produce a number of different cost estimates based on how it is undertaken and what assumptions are made. This reinforces the need for sensitivity analyses and a well-defined “systems” approach.
We divide the techniques of costing into three main categories: revealed preference, stated preference, and implied preference. Revealed preference is based on observed conditions and how individuals subject to the externality behave, stated preference comes from surveys of individuals in hypothetical situations, while implied preference looks at the cost which is implied based on legislative, executive, or judicial decisions.

### Revealed Preference

The revealed preference approach attempts to determine the cost of an externality by determining how much damage reduces the price of a good.

Revealed preference can also be used to estimate the price people pay for various protection (defense/ abatement) measures and the effectiveness of those measures. For instance, insulation costs a certain amount of money and provides a certain amount of effectiveness in reducing noise. The extent to which individuals then purchase insulation or double-glazed windows may suggest how much they value quiet. However, individuals may be willing to spend some money (but less than the cost of insulation) if they could ensure quiet by some other means which they do not control – but which may be technically feasible.

Hedonic Models: The most widely used estimates of the cost of noise are derived from hedonic models. These assume that the price of a good (for instance a home) is composed of a number of factors: square footage, accessibility, lot area, age of home, pollution, noise, etc. Using a regression analysis, the parameters for each of these factors are estimated. From this, the decline in the value of housing with the increase in the amount of noise can be estimated. This has been done widely for estimating the social cost of road noise and airport noise on individual homes. In theory, the value of commercial real estate may be similarly influenced by noise. In our literature review thus far, no study of this sort has been found. Furthermore, although noise impacts public buildings, this method cannot be used as a measure since public buildings are not sold. Similarly, when determining some of the costs of noise, one could investigate how much individuals might be willing to pay for vehicles which are quieter. Like a home, a hedonic model of vehicle attributes could be estimated. A vehicle is a bundle of attributes (room, acceleration, MPG, smooth ride, quiet, quality of workmanship, accessories) which influence its price, also an attribute.

Unit/Cost Approach: A simple method, the “unit cost (Rate) approach” is used often for allocating costs in transit. This method assigns each cost element, somewhat arbitrarily, to a single output measure or cost center (for instance, Vehicle Miles Travel, Vehicle Hours Travel, Number of Vehicles, Number of Passengers) based on the highest statistical correlation of the cost with output.

Wage/Risk Study: A means for determining the economic cost of risk to life or health or general discomfort is by analyzing wage/salary differentials based on job characteristics, including risk as a factor.

Time Use Study: This approach measures the time used to reduce some risk by a certain amount. For instance, seatbelts reduce the risk of injury or using pedestrian overpass may reduce the risk of being hit by a car. The time saved has a value, which may inform estimates of risk aversion.

Years Lost plus Direct Cost: This method estimates the number of years lost to an accident due to death and years lost from non-fatal injuries. It also the monetary costs of non-life damages. However, it defines life in monetary terms. While it may have some humanistic advantages in that it does not place a dollar value on life, defining life through dollars and sense may have some practical value. Defining life through dollars and sense may help us assess whether an improvement, with a certain construction cost and life-saving potential, is economically worthwhile.

Comprehensive: This accident costing method extends the Years Lost plus Direct Cost method by placing a value on human life. The value is assessed looking at the tradeoffs people make when choosing to conduct an activity a certain risk level versus another activity at a different risk, but different cost/time. Studies are based both on what people actually pay and what are willing to pay, and use a variety of revealed preference techniques. This is the preferred method of the US Federal Highway Administration.

Human Capital: The Human Capital approach is an accounting approach which focuses on the accident victim’s productive capacity or potential output, using the discounted present value of future earnings. To this are added costs such as property damage and medical costs. Pain and suffering can added as well. The Human Capital approach can be used for accidents, environmental health, and possibly congestion costs . It is used in the Australian study Social Cost of Road Accidents (1990) [19]. However, Miller (1992) [20] and others discount the method because the only effect of injury that counts is the out-of-pocket cost plus lost work and housework. By extension, it places low value on children and perhaps even a negative value on the elderly. While measuring human capital is a necessary input to the costs of accidents, it cannot be the only input.

### Stated Preference

Stated preference involves using hypothetical questions to determine individual preferences regarding the economic costs of a facility. There are two primary classes of stated preference studies: Contingent Valuation and Conjoint Analysis.

Contingent Valuation: Perhaps the most straight-forward way of determining the cost of an externality is asking the hypothetical questions, “How much you would a person pay to reduce externality by a certain amount” or “How would a person pay to avoid the imposition of a certain increment of externality”. Jones-Lee (1990) [21] has been the foremost investigator into this method for determining the cost of noise. This method can, in theory, be applied to any recipient of noise, although it has generally been asked of the neighbors (or potential neighbors) of a transportation facility.There are several difficulties with this approach. The first difficulty with any stated preference approach is that people give hypothetical answers to hypothetical questions. Therefore, the method should be calibrated to a revealed preference approach (with actual results for similar situations) before being relied upon as a sole source of information. The second regards the question of “rights”. For instance, someone who believes he has the right to quiet will not answer this question in the same way as someone who doesn’t. The third involves individuals who may claim infinite value to some commodity, which imposes difficulties for economic analysis.

Conjoint Analysis: To overcome the problems with contingent valuation, conjoint analysis has been used. Conjoint analysis requires individuals to tradeoffs between one good (e.g. quiet) and another (e.g. accessibility) has been used to better measure the cost of noise, as in Toronto by Gillen (1990) [22].

### Implied Preference

There are methods for measuring the costs of externalities which are neither revealed from individual decisions nor stated by individuals on a survey. These are called implied preference because they are derived from regulatory or court-derived costs.

Regulatory Cost: Through government regulation, costs are imposed society with the aim of reducing the amount of noise or pollution or hazard is produced. These regulations include vehicle standards (e.g. mufflers) roadway abatement measures such as noise walls, as well as the many environmental regulations. By determining the costs and benefits of these regulations, the implicit cost of each externality can be estimated. This measure assumes that government is behaving consistently and rationally when imposing various standards or undertaking different projects.

Judicial Opinion and Negotiated Compensation: Similar to the implicit cost measure, one can look at how courts (judges and juries) weigh costs and benefits in cases which come before them. The cost per unit of noise or life from these judgments can be determined. This method is probably more viable in accident cases.

## Incidence, Cost Allocation, and Compensation

This final set of topics deal with incidence (who causes the externality), cost allocation (who suffers from the externality), and compensation (how can the costs be appropriated and compensation paid fairly).

### Incidence

The general model is that the costs can be generated by one of several parties and fall on one of several parties. The parties in this case are: the vehicle operators and carriers; the road, track, and airport operators; and the rest of society.

• Vehicle Operators and Carriers: bus company, truck company, driver of a car, railroad, airline
• Society: the citizenry, government, citizens of other states/countries, the environment

This conceptual model is not concerned with anything smaller than the level of a vehicle. How costs on a vehicle are attributed to passengers in the vehicle, or the costs of freight carriage to the shipper, is not our concern. Similarly, ownership is not an issue, the operator of a vehicle may not be the owner, in the case, for instance, of a rented car. Obviously there is some overlap here between vehicle operators and road and track operators. In the case of American railroads, the firm which operates trains usually owns the track, although often a train will ride on tracks owned by a different railroad. Moreover, for some means of transportation, but not those considered here, there may be no vehicles (for example pipelines and conveyor belts.)

Costs can be imposed by any party on any party. As an illustration of how this works, we look at noise. Transportation noise is generated by vehicles in motion, and can affect any of the following classes: self, other vehicle users, and local society. There is noise generated by the roadway or the rail during construction, but this is ignored, and the noise does not actually hurt the road and track operators (except indirectly where they are held responsible for noise generated by vehicles and must build noise walls or other abatement measures.) A similar situation occurs with airports. Technically the planes make almost all of the noise, but the airport is held responsible. That noise is generated by wheels on pavement and thus depends in some respects on the roadway operator is also ignored.

• Vehicle operator on self, on other vehicles. For instance, one of the attributes of a vehicle (an auto say) is its quietness, this is reflected in the price of the vehicle. Quietness has two aspects: insulation, which protects the cab from noise generated by the car and other vehicles; and noise generation, which is how noisy the car is to itself and others. The noise generated by the vehicle and heard within the cab are internal costs, while those generated by the vehicle and heard by others is external to the vehicle operator, but internal to the transportation system.
• Vehicle operator on society. The noise generated by a vehicle negatively impacts the usefulness and flexibility of land uses nearby, where the impact declines with distance. The decline in utility is reflected in land values. The costs are clearly external to both the operator and the transportation system.

### Cost Allocation

Clearly there are external costs, but it is not always clear who should bear them. This issue brings about questions of cost allocation. These include: objectives – for what reason are we allocating costs, methodology – how are we allocating costs, structure – how do we break down costs, and problems – how do we deal with the thorny issues of common and joint costs and cross-subsidies.

The first question that must be asked is what are the objectives of cost allocation. There are several contenders, which unfortunately are not entirely compatible. These include equity, efficiency, effectiveness, and acceptability.

The first consideration is equity or fairness. This concept raises a series of question summarized as “equity for whom”. Depending on how you slice it, different “fair” solutions are possible. The classic divisions are vertical vs. horizontal equity. Horizontal equity is a fair allocation of costs between users in the same sector, vertical equity is fairness across sectors. Are the costs allocated “fairly” between users, between facilities, between modes, between economic sectors? Is the burden for the project shared fairly between the economy and the environment?
The second consideration is efficiency. Somewhat clearer than equity, efficiency still raises the same questions of “for whom.” Is the allocation efficient for the user, the operator, the state, the country? Does it consider inefficiencies, subsidies and taxes in other sectors of the economy, or other components of the transportation system? Efficiency can also be stratified into two categories: theoretical and practical.. The first ignores implementation (information and transaction) costs that rise with the number of charges imposed. Moreover, economists identify three kinds of efficiency: Allocative, which aims for the optimal mix of goods; Productive, which attempts to attain the minimum average cost; and Dynamic, which seeks long term optimal investment or capital rationing. Allocative efficiency may be thought of as congestion pricing, to ensure the optimal use of a transportation facility. Productive efficiency will attempt to raise enough money to operate and maintain the physical plant at the lowest cost. Dynamic efficiency will attempt to raise money to finance the facility, proactively or retroactively. To what extent these goals coincide is unclear.

Contrasted with efficiency is effectiveness. While the test of efficiency asks if the system is achieving its goals with minimum effort, the test of effectiveness asks if the system’s goals or output measures are consistent with broader societal goals. For instance, an efficient road may move traffic through a neighborhood at a high rate of speed, but this may be ineffective in meeting the broader social goal of a higher quality of life in the neighborhood, which the traffic disrupts. Costs can be allocated which achieve an efficient use of resources, but result in an ineffective or counter-productive system.

Added to this, we will consider the profit motive. If the facility is constructed by a profit seeking firm, prices will reflect an attempt at profit maximization in either a competitive, monopolistic, or oligopolistic environment.

A last consideration is acceptability. A system, which may have desirable attributes, if unimplemented, serves no-one. In the political world, tradeoffs and compromises must be made to achieve progress.

Costs can be allocated based on who causes them or by who receives benefit from them. There are pricing schemes reflecting both. There is a dichotomy between the methods of cost allocation suggested by economists and the approaches taken by engineers (as well as the official policy of the US government through modal cost allocation studies).

At least three economic approaches can be taken for allocating costs. The economic top-down approaches take equations of cost and allocate the results to users, these are: average total cost per user, average variable cost per user, and marginal cost (short run and long run), the last of which is favored by economists.

On the other hand, engineers working from the bottom-up break the system into components, which are assigned to users. Each mode or carrier has somewhat different methods for cost allocation. These are summarized below:

• Fixed Allocation – a set fee is charged based on some previous study
• Industry Agreed Upon (e.g. General Managers Associations Rules – rules allocating costs of freight cars on foreign rails, a pre-established agreement)
• Zero Allocation – user gets free ride on common costs and pays only attributable costs
• Proportional (New Investment/Long Range Pricing) – divides variable and fixed costs to users in proportion to use
• Minimum Cost of Service: Avoidable Cost Allocation (hierarchy costs/avoidable costs/separable costs/remaining benefits) – assigns to a beneficiary only the costs which could be avoided if the beneficiary did not use the service
• Minimum Cost of Service: Attributable Cost Allocation – assigns as cost allocation + share of common costs based on use.
• Minimum Cost of Service: Priority of Use Cost Allocation – assigns attributable cost allocation, but charges extra if priority is given to user or discounts if priority is taken from user (e.g. queue jumping)

In addition to the centralized cost allocation methods described above, there are other methods of allocation to users:

• Negotiated contracts – the parties negotiate the charge based on individual circumstances. This is often used in the rail industry where the trains of one carrier use the tracks of another.
• Arbitration – like a negotiated contract, but where a third party makes ultimate decision on the charge.
• Regulatory finding – A regulatory agency such as the former Interstate Commerce Commission gathers information and makes a decision as to appropriate rate. This is now most widely used in cases of monopoly oligopoly practice.
• Legislative finding – A legislature assumes the role of regulatory agency and prices and/or conditions of the cost allocation. An example of this is the adoption of taxes supporting the highway system, where gas taxes, vehicle licenses, and truck charges as well as tolls have to be approved by the state legislature.
• Judicial finding – After some dispute between parties (carrier vs. carrier, carrier vs. government or government vs. government) a court may be called on to make a final decision.
• Ramsey Pricing Rule – This rule would charge based on the customer’s elasticity of demand. The more elastic the customer (the more options he has the lower his price. So long as the short run marginal cost is covered, it may worthwhile for one firm to use this pricing rule to keep customers using their service rather than a competitors.
• Discriminating Monopolist/Oligopolist An unregulated monopoly discriminate among customers to obtain higher revenues (capture the consumer surplus). There are three classes of monopolistic discrimination: (1st degree, degree, 3rd degree).

The engineering and economic cost allocation discussed above allocate the costs to users. But there are alternative approaches:

• General Revenue: If transportation is to be subsidized, then the general public (including both users and non-users) can be charged a certain percentage of costs. This is seen when using general tax revenue for transportation.
• Value Capture: Similarly, another transfer occasionally used is a “value capture” approach, whereby nearby landowners are taxed based on the increase property value owing to a new transportation facility, this has been used in Angeles around new transit stations. In practice, some of each approach may be used.

### Compensation

If individuals and organizations who cause externalities are to be charged, those who receive the unwanted noise, pollution, etc. should be compensated. To the extent that the recipients are amorphous, such as the environment, the collected funds should be expended in that sector for remediation of damages or their mitigation ahead of time. Also, the health damages from environmental damage are typically diffuse. On the other hand, it is fairly clear who suffers from noise. But the externality gets buried in the land price immediately after the opening (or perhaps announcement) of a facility. Therefore only the land owner at that time should receive compensation.
Crashes result in damages to several classes of parties: those involved in crashes (and their families and insurance companies), commuters delayed by crashes (though this may be better treated in the congestion section), and society at large. Those involved are largely covered privately through the insurance sector, and care must be taken to avoid double-counting.

Congestion is typically divided into two classes: recurring and non-recurring. Non-recurring congestion is most often caused by incidents (traffic accidents, inclement weather). The value of time for these may be different, as recurring congestion probably entails less schedule delay since it is already accounted for by most commuters. Money raised from congestion pricing, in addition to reducing traffic volumes, can be used to expand capacity further to alleviate congestion. But this does not compensate those who now take a slower (but cheaper mode of transport) after road pricing is in effect. A question arises as to whether those individuals have some right to free travel which is being eliminated through pricing, or whether some general subsidies for travel are warranted. Congestion has further issues concerning pricing, for instance the peak vs. off-peak. When there is more traffic, each additional vehicle has more and more impact, suggesting higher tolls in the peak. However, the tolls will reduce demand, so an equilibrium solution to the problem is essential.

Social severance and visual impact are also amorphous. They will be difficult to price. To some extent for visual impact, the neighbors of a project can be identified and damages defined in terms of lower property values. In terms of the aesthetic quality of a trip, it may be conceptually possible to compare to parallel routes (a parkway vs. a freeway), one prettier than the other, and see if there is a difference in traffic volumes other than that explained by a route choice model. The difference in volume gives an implied choice of the value of the route in terms of additional time (and thus money), which may be significant in tourist areas. There is also a risk aspect to travel, drivers may choose certain roads which are through good areas, because they do not want to break down in isolated areas or perceived bad neighborhoods.

The social aspects of disruption of community (after taking into account net change in property value before and after infrastructure accounting for all of accessibility (increase or decrease), noise, and visual impact) is extremely difficult to determine. A political solution may need to be found to pricing and arranging for compensation.

## Evidence on Pollution

Transportation sources in North America contribute approximately

• 47% of nitrogen oxide emissions (NOx)
• 71% of carbon monoxide emissions (CO)
• 39% of hydrocarbon emissions (HC)

## Standards vs. Prices

To control most pollutants we have opted for standards rather than pricing. This is reflected in the ‘level of allowed emissions’ with catalytic converters on our vehicles.

Noise is another example where the U.S. has opted for a technological fix to achieve a standard. Europeans have, however, introduced noise charges at some airports for aircraft which exceed a particular noise level.

## Private Cost v. Social Cost

The purpose of distinguishing private and social cost is to correct for real resource misallocation from economic agents actions which impose a cost (or benefit) on others in the market. The market provides no incentive for agents to take account of their actions.

The difference between private and social cost is that in making a decision a private individual will take account of the costs they face but will not consider the impact of their decision on others which may, in fact. impose a cost upon them. If this occurs an externality will misallocate resources since the economic agents are not forced to pay the cost they impose or does not receive any compensation for the benefits which they confer.

## Full Cost Model

Based on: [23]

An essential first step in examining transportation issues and in making sound decisions on transportation systems is to understand the full cost of transportation today, including the social costs of crashes, air pollution, noise, and congestion as well as the internal costs of providing and operating the infrastructure. Furthermore, if cross subsidies between modes, user groups, or areas of the country or states are to be avoided, and if users are to pay the full cost of providing and maintaining the transportation system, then it is important to know what proportion of total costs users currently pay and what proportion is borne by others. Such a complete assessment of the full cost of the different modes of transportation for intercity travel has been lacking. The development of cost models and estimates of the type presented in this research are essential to gauging the true costs of transportation in the different modes, and is a prerequisite to sound investment decisions.

The full cost calculation includes the cost of building, operating, and maintaining infrastructure, as well as carrier, user, and social costs. Social costs include noise, air pollution, and accident costs, as well as congestion costs. User costs include the cost of purchasing, maintaining and operating a vehicle such as a car, and the cost of travel time.
We begin by developing a taxonomy for representing the full costs of transportation, independent of mode:

• Infrastructure Costs
${displaystyle C_{I}}$

– including capital costs of construction and debt service, and costs of maintenance and operating costs as well as service costs to government or private sector;

• Carrier Costs – aggregate of all payments by carriers in capital costs to purchase a vehicle fleet), and maintain and operate a vehicle fleet (COC), minus those costs (such as usage charges) which are transfers to infrastructure, which we label Carrier Transfers.
• User Money Costs
${displaystyle C_{U}}$

– aggregate of all fees, fares and tariffs paid by users in capital costs to purchase a vehicle, and money spent to maintain and operate the vehicle or to ride on a carrier (UOC); less those costs (such as fares) which are transfers to carriers or infrastructure, and accident insurance, which is considered under social costs, which we label User Transfers

${displaystyle T_{U}}$

.

• User Travel Time Costs
${displaystyle C_{T}}$

– the amount of time spent traveling under uncongested conditions multiplied by the monetary value of time.

• User Delay Costs
${displaystyle C_{D}}$

– the amount of time spent traveling under congested conditions minus the amount of time spent traveling in uncongested conditions multiplied by the monetary value of time.

• Social Costs – additional net external costs to society due to emissions
${displaystyle C_{E}}$

, crashes

${displaystyle C_{A}}$

, and noise

${displaystyle C_{N}}$

and are true resource costs used in making and using transportation services;

The method used to estimate the full cost

${displaystyle C_{Full}}$

of intercity travel will combine elements from a number of sources. Adding and subtracting the above factors, thereby avoiding double-counting, we have the following equation, the components of which will be dealt with in turn in the paper:

${displaystyle C_{Full}=(C_{U}-T_{U})+C_{I}+C_{E}+C_{N}+C_{A}+C_{T}}$

### Key Issues

“Externalities” are Inputs to Production System. Clean Air, Quiet, Safety, Freeflow Time are used to produce a trip. The System has boundaries: Direct effects vs. Indirect effects Double Counting must be avoided

### Selection of Externalities

Criteria: Direct Effects

Not Internalized in Capital or Operating Costs

External to User (not necessarily to system)

Result: Noise, Air Pollution, Congestion, Crashes

Not: Water Pollution, Parking, Defense …

### Noise

#### Measurement

Noise: Unwanted Sound

dB(A) = 10 log (P2/Pref)

P: Pressure, Pref: queitest audible sound

NEF: Noise Exposure Forecast is a function of number (frequency) of events and their loudness.

#### Generation

Amount of noise generated is a function of traffic flow, speed, types of traffic.

Additional vehicles have non-linear effect: e.g. 1 truck = 80 db, 2 trucks = 83 db, but sensitivity to loudness also rises

Noise decays with distance

#### Valuation

Hedonic Models: Decline of Property Values with Increase in Noise –> Noise Depreciation Index (NDI).
Average NDI from many highway and airport studies is 0.62. For each unit increase in dB(A), there is a 0.62% decline in the price of a house

#### Integration

Noise Cost Functions ($/pkt) : f(Quantity of Noise, House Values, Housing Density, Interest Rates) Using “reasonable” assumption, this ranges from$0.0001/vkt – $0.0060/vkt for highway. Best guess =$0.0045/pkt.

#### Integration

Time Cost Functions:

TC = VoT Qh ( Lf/ Vf + a (Qh / Qho)b)

highway: a=0.32, b=10

air: a=2.33, b=6

### Crashes

#### Measurement

Number of Crashes by Severity
Multiple Databases (NASS, FARS)
Multiple Agencies (NHTSA, NTSB), + states and insurance agencies
Inconsistent Classification
Non-reporting

#### Generation

Crash Rates, Functions
Highway: Crash Rate = f(urban/rural, onramps, auxiliarly lanes, flow, queueing)
Air: Crash Rate = f( type of aircraft)

#### Valuation

The principal means for estimating the cost of crashes is to estimate their damage costs. The method presented here uses a comprehensive approach which includes valuing years lost to the accident as well as direct costs. Several steps must be undertaken: converting injuries to years of life, developing a value of life, and estimating other costs. Placing a value on injury requires measuring its severity. Miller (1993) describes a year of functional capacity (365 days/year, 24 hours/day) as consisting of several dimensions: Mobility, Cognitive, Self Care, Sensory, Cosmetic, Pain, Ability to perform household responsibilities, and Ability to perform wage work. The following Tables show the percent of hours lost by degree of injury, and the functional years lost by degree of injury.

Percentage of Hours Lost to Injuries by Degree of Injury

 Type of Activity Modest Major Fatal Total Functioning 18.0 40.7 41.3 100.0 HH Production 25.2 22.1 52.7 100.0 Work 21.7 19.1 59.2 100.0

source Miller (1991) p.26

Functional Years lost by Degree of Injury

 Degree of Injury Per Injury Percent of Lifespan Per Year Percent of Annual Total 1. Minor 0.07 0.15 316,600 10.7 2. Moderate 1.1 2.3 587,700 20.0 3.Serious 6.5 13.8 1,176,700 40.0 4. Severe 16.5 35.0 446,700 15.2 5. Critical 33.1 70.0 413,800 14.1 Avg. Nonfatal 0.7 1.5 2,941,500 100.0 Fatal 42.7 100.0 2.007,000

source Miller (1991) p29
note: expected lifespan for nonfatally injured averages 47.2 years

Federal Highway Administration uses the following:

 MAIS Level Severity Fraction of VSL MAIS 1 Minor 0.0020 MAIS 2 Moderate 0.0155 MAIS 3 Serious 0.0575 MAIS 4 Severe 0.1875 MAIS 5 Critical 0.7625 MAIS 6 Fatal 1.0000

source:[24]

Central to the estimation of costs is an estimate of the value of life (or value of a statistical life). Numerous studies have approached this question from various angles. Jones-Lee (1988) provides one summary, with an emphasis on British values from revealed and stated preference studies. The FAA (1989) provides another summary. He finds the range of value of life to vary by up to two orders of magnitude (a factor of 100). Miller’s (1991) summary is reproduced below, with numbers updated to 1995 dollars.

Estimated Value of Life by Type of Study

 Type of Study Value of Life ($) (1988 dollars) Value of Life ($) (1995 dollars) Average of 49 studies 2.2 M 2.9 M Average of 11 auto safety studies 2.1 M 2.7 M Study Type Extra wages for risky jobs (30 studies) 1.9-3.4 M 2.5 – 4.4 M Market demand vs. price safer cars 2.6 M 3.4 M smoke detectors 1.2 M 1.6 M houses in less polluted areas 2.6 M 3.4 M life insurance 3.0 M 3.9 M wages 2.1 M 2.7 M Safety behavior pedestrian tunnel use 2.1 M 2.7 M safety belt use (2 studies) 2.0 – 3.1 M 2.6 – 4.0 M speed choice (2 studies) 1.3 -2.2 M 1.7 – 2.9 M smoking 1.0 M 1.3 M Surveys Auto safety (5 studies) 1.2-2.8 M 1.6 – 3.6 M Cancer 2.6 M 3.4 M Safer Job 2.2 M 2.9 M Fire Safety 3.6 M 4.7 M

Source: Miller (1990),
Note: in millions (M) of after-tax dollars ($1995 =$1988 * 1.3).

Currently (as of 2008 $) FHWA uses$5.8 M[25], which is the average of several recent studies.

#### Integration

Highway Accident Costs estimates range from $0.002 –$0.09/pkt. Our estimate is $0.02/pkt. Urban / rural tradeoff. Urban more but less severe crashes. Air Accident Costs$0.0005/pkt.

### Summary

Costs in $per passenger km traveled.  Cost Category Air System Highway System Noise$0.0043 $0.0045 Air Pollution$0.0009 $0.0031 Crashes$0.0004 $0.0200 Congestion$0.0017 $0.0046 TOTAL$0.01 0.03 High Uncertainty About Valuation Costs Vary with Usage Accounting, Difficult, but necessary to avoid double counting. ## Thought Question: Value of Life Suppose there is a road improvement which will save 1 life per year, reducing the number of fatalities from 2 to 1 per year (out of 1000 people using the road). Assume all travelers are identical. What value of life should be used in the analysis? Normally, we would do the equivalent of trying to compute for each traveler what is the willingness to pay for a 50% reduction in the chance of death by driving (from 2 in 1000 to 1 in 1000), and multiply that by the 1000 people whose chance of dying is reduced. An alternative approach is to figure out the willingness to pay for the driver whose life is saved. So how much would you pay to avoid dying (with certainty) (i.e. what is your Willingness to Pay)? The answer to the first question is usually taken to be all of your resources (you would pay you everything so I won’t kill you). Alternatively how much can I pay you to allow you to let me kill you (Willingness to Accept)? The answer to this second question is: I would have to pay you an infinite amount of money in order for you to let me kill you. Both of those sums of money (everything or infinity) likely exceed the willingness to pay to reduce the likelihood of dying with some probability, multiplied by the number of people experiencing it. In economic terms, we are comparing the area under the demand curve (the consumer’s surplus) for life (which has a value asymptotically approaching infinity as the amount of life approaches 0 (death approaches certainty) for a single individual, with the marginal change in the likelihood of survival multiplied by all individuals (i.e. the quadrilateral between the y-axis of price and the same demand curve, between Pb and Pa) which describes the change in price for a change in survival). On the one hand, using the marginal change for everyone rather than total change for the one person whose life is saved, we will give a lower value to safety improvements. On the other hand, the value of life to the individual himself is much higher than the value of life of that individual to society at large. ## References 1. Keeler, T.E., K. 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IBI Group (1995) Full Cost Transportation Pricing Study: Final Report to Transportation and Climate Change Collaborative. 8. Verhoef, Erik External Effects and Social Costs of Road Transport Transportation Research A Vol. 28A No. 4 pp. 273-387, 1994 9. Rothengatter, Werner Do External Benefits Compensate for External Costs of Transport Transportation Research A Vol. 28A No. 4 pp. 321-328, 1994 10. Button, Kenneth Alternative Approaches Toward Containing Transport Externalities: An International Comparison Transportation Research A Vol. 28A No. 4 PP. 289-305, 1994 11. Gwilliam, Kenneth M. and Harry Geerlings New Technologies and Their Potential to Reduce the Environmental Impact of Transportation Transportation Research A Vol. 28A No. 4 pp. 307-319, 1994 12. Rietveld, Piet Spatial Economic Impacts of Transport Infrastructure Supply Transportation Research A Vol. 28A No. 4 pp. 329-341, 1994 13. Coase, R.H. The Problem of Social Cost, and Notes on the Problem of Social Cost reprinted in The Firm, The Market and the Law University of Chicago Press (1992) 14. Stigler, George 1966 The Theory of Price, 3rd ed. (New York: Macmillan and Col. 1966), 113 15. Pigou, A.C. 1920 orig. The Economics of Welfare, 4th ed. (London, Macmillan and Co. 1932) 16. Button, Kenneth Alternative Approaches Toward Containing Transport Externalities: An International Comparison Transportation Research A Vol. 28A No. 4 PP. 289-305, 1994 17. DeLuchi, M.A. (1991) Emissions of Greenhouse Gases from the Use of Transportation Fuels and Electricity: Volume 1 Main Text. Department of Energy: Argonne National Laboratory, Center for Transportation Research, Energy Systems Division. 18. Becker, Gary, A Theory of the Allocation of Time – The economic journal, 1965 pp. 493–517 19. Australian Bureau of Transport and Communications (1992) Social Cost of Road Accidents in Australia, Economics Report 79, Australian Government Publishing Service, Canberra, Australia 20. Miller, Ted (1992) The Costs of Highway Crashes, Federal Highway Administration (FHWA-RD-91-055) 21. Jones-Lee, Michael The Value of Transport Safety. Oxford Review of Economic Policy, Vol. 6, No. 2, Summer 1990 pp. 39-60 22. Gillen, David (1990) The Management of Airport Noise, DWG Research Associates for Transport Development Centre, Transport Canada, July 1990 23. Levinson, David, David Gillen, and Adib Kanafani (1998) A Comparison of the Social Costs of Air and Highway. Transport Reviews 18:3 215-240.) 24. Duvall, Tyler (2008) Treatment of the Economic Value of a Statistical Life in Departmental Analyses http://ostpxweb.dot.gov/policy/reports/080205.htm 25. http://www.fhwa.dot.gov/policy/2008cpr/appb.htm See also: Utility Applied to Mode Choice (Fundamentals of Transportation wikibook) Utility is the economists representation of whatever consumers try to maximize. Consumers may want more of one thing and less of another. … ## Indifference Curves Demand depends on utility. Utility functions represent a way of assigning rankings to different bundles such that more preferred bundles are ranked higher than less preferred bundles. A utility function can be represented in a general way as: ${displaystyle U=U(x_{1},x_{2})=x_{1}x_{2},!}$ where ${displaystyle x_{1}}$ and ${displaystyle x_{2}}$ are goods (e.g. the net benefits resulting from a trip) An indifference curve is the locus of commodity bundles over which a consumer is indifferent. If preferences satisfy the usual regularity conditions (discussed below), then there is a utility function ${displaystyle U(x_{1},x_{2})}$ that represents these preferences. Points along the indifference curve represent iso-utility. The negative slope indicates the marginal rate of substitution (MRS): ${displaystyle MRS=-{frac {Delta x_{2}}{Delta x_{1}}},!}$ ## Substitutes and Complements Substitutes would be represented by : ${displaystyle U(x_{1},x_{2})=ax_{1}+bx_{2},!}$ where the slope of the indifference curve would be = -a/b. In graphic terms substitutability is greater the more the indifference curves approach a straight line. Perfect substitutability is a straight line indifference curve (e.g. trips to work by mode A or mode B). Complements are represented by: ${displaystyle U(x_{1},x_{2})=min(x_{1},x_{2}),!}$ The more complementary the more the indifference curves approach a right angle curve; perfect complementarity would have a right angle indifference curve (eg. left and right shoes, trips from home to work and work to home) ## Trade Game The trade game is a way of examining how economic trading of resources affects individual utility. Imagine the economy consists of the following resources (denoted by colored slips of paper) • White • Purple • Brown • Orange • Blue • Gray • Green • Yellow • Gold The objective of the game is to maximize your gains in utility. Define A Utility Function for Yourself ${displaystyle U=f({text{White, Purple, Brown, Orange, Blue, Gray, Green, Yellow, Gold}})}$ You are handed an assortment of resources Measure your utility Trade with others in the class (15 minutes) Record your trades. At the end of the trading period measure your utility again. Compute your absolute and percentage increase. Record scores on the board Discuss Is there a better way to allocate resources? ## Preference Maximization ### Graphical Budget curve moves outward Slope of budget curve changes Utility maximization involves the choice of bundles under a resource constraint. For example, individuals select the amount of goods, services and transportation by comparing the utility increase with an increase in consumption against the utility loss associated with the giving up of resources (or equivalently forgoing the consumption which those resources command). Often one price is taken to be 1, and one good is taken to be money. An income increase can be represented by the outward movement of the budget line. An increase in the price of good ${displaystyle X_{1}}$ can be represented by a change in the slope of the budget line (still anchored at one end). In graphic terms the process of optimization is accomplished by equating the rate at which an individual is willing to trade off one good for another to the rate at which the market allows him/her to trade them off. This can be represented in the following graph The individual maximizes utility by moving down the budget constraint to that point at which the slope of the budget line ( ${displaystyle -P_{1}/P_{2}}$ ) which is the rate of exchange dictated by the market is just equal to the rate at which the individual is willing to trade the two goods off. This is the slope of the indifference curve or the marginal rate of transformation (MRT). A point such as ‘e’ is an equilibrium point at which utility is being maximized. Equilibrium is the tangency between the indifference curve/utility and the budget constraint. ### Optimization As an optimization problem, this can be written: ${displaystyle {text{Maximize}}U(X),!}$ subject to: ${displaystyle pxleq m,!}$ ${displaystyle x}$ is in ${displaystyle X}$ where: • ${displaystyle p}$ = price vector, • ${displaystyle x}$ = goods vector, • ${displaystyle m}$ = income (Because of non-satiation, the constraint can be written as px=m.) This kind of problem can be solved with the use of the Lagrangian: ${displaystyle Lambda =U(X)-lambda (px-m),!}$ where • ${displaystyle lambda }$ is the Lagrange multiplier Take derivatives with respect to ${displaystyle x}$ , and set the first order conditions to 0 ${displaystyle {frac {partial Lambda }{partial x_{i}}}={frac {partial U(X)}{partial x_{i}}}-lambda p_{i}=0,!}$ Divide to get the Marginal rate of substitution and Economic Rate of Substitution ${displaystyle MRS={frac {frac {partial U(X)}{partial x_{i}}}{frac {partial U(X)}{partial x_{j}}}}={frac {p_{i}}{p_{j}}}=ERS,!}$ ### Example: Optimizing Utility ${displaystyle {text{Max}}U=x_{1}x_{2},!}$ ${displaystyle {text{s.t.}}m=p_{1}x_{1}+p_{2}x_{2},!}$ ${displaystyle Lambda =x_{1}x_{2}-lambda (p_{1}x_{1}+p_{2}x_{2}-m),!}$ ${displaystyle {frac {partial Lambda }{partial x_{1}}}=x_{2}-lambda p_{1}=0,!}$ ${displaystyle {frac {partial Lambda }{partial x_{2}}}=x_{1}-lambda p_{2}=0,!}$ Solving ${displaystyle {frac {x_{2}}{p_{1}}}={frac {x_{1}}{p_{2}}},!}$ or ${displaystyle p_{1}x_{1}{text{ }}={text{ }}p_{2}x_{2},!}$ substituting into the budget constraint: ${displaystyle m=2p_{1}x_{1},!}$ ${displaystyle x_{_{1}}^{*}={frac {m}{2p_{1}}},!}$ ${displaystyle x_{_{2}}^{*}={frac {m}{2p_{2}}},!}$ ## Demand, Expenditure, and Utility ### Indirect Utility The Marshallian Demand relates price and income to the demanded bundle. This is given as ${displaystyle x(p,m)}$ . This function is homogenous of degree 0, so if we double both ${displaystyle p}$ and ${displaystyle m}$ , ${displaystyle x}$ remains constant. We can develop an indirect utility function: ${displaystyle v(p,m)=maxU(X),!}$ subject to: ${displaystyle px=m,!}$ where X that solves this is the demanded bundle #### Example: Indirect Utility ${displaystyle left({frac {m}{{text{2}}p_{1}}}right)left({frac {m}{{text{2}}p_{2}}}right)={frac {m^{2}}{4p_{1}p_{2}}},!}$ taking a monotonic transform: ${displaystyle =left({frac {1}{4}}right)left(2ln left(mright)-ln left(p_{1}right)-ln left(p_{2}right)right),!}$ which increases in income and decreases in price where the ${displaystyle X}$ that solves this is the demanded bundle #### Properties Properties of the indirect utility function ${displaystyle v(p,m)}$ • is non-increasing in ${displaystyle p}$ , non-decreasing in ${displaystyle m}$ • homogenous of degree 0 • quasiconvex in ${displaystyle p}$ • continuous at all ${displaystyle p>>0,m>0}$ Expenditure Function The inverse of the indirect utility is the expenditure function ${displaystyle e(p,u)={text{min}}px,!}$ subject to: ${displaystyle u(x)geq u,!}$ Properties of the expenditure function ${displaystyle e(p,u)}$ : ### Roy’s Identity The Hicksian Demand or compensated demand is denoted h(p,u). ${displaystyle h_{i}(p,u)={frac {partial e(p,u)}{partial p_{i}}},!}$ vary price and income to keep consumer at fixed utility level vs. Marshallian demand. Roy’s Identity allows going back and forth between observed demand and utility ${displaystyle x_{i}(p,m)=-{frac {frac {partial v(p,m)}{partial p_{i}}}{frac {partial v(p,m)}{partial m}}},!}$ #### Example (continued) ${displaystyle x_{1}(p,m)=-{frac {{}^{partial V}!!diagup !!{}_{partial p};}{{}^{partial V}!!diagup !!{}_{partial m};}}=-{frac {-{}^{m^{2}}!!diagup !!{}_{4p_{1}^{2}p_{2}^{}};}{{}^{2m}!!diagup !!{}_{4p_{1}^{}p_{2}^{}};}}={frac {2m}{p_{1}}},!}$ ${displaystyle x_{2}(p,m)={frac {2m}{p_{2}}},!}$ ### Equivalencies ${displaystyle e(p,v(p,m))=m,!}$ the minimum expenditure to reach ${displaystyle v(p,m)}$ is ${displaystyle m}$ ${displaystyle v(p,e(p,u))=u,!}$ the maximum utility from income ${displaystyle e(p,u)}$ is ${displaystyle u}$ ${displaystyle x_{i}(p,m)=h_{i}(p,v(p,m)),!}$ Marshallian demand at ${displaystyle m}$ is Hicksian demand at ${displaystyle v(p,m)}$ ${displaystyle h_{i}(p,u)=x_{i}(p,e(p,u)),!}$ Hicksian demand at ${displaystyle u}$ is Marshallian demand at ${displaystyle e(p,u)}$ ## Measuring Welfare ### Money Metric Indirect Utility Function The Money Metric Indirect Utility Function tells how much money at price p is required to be as well off as at price level q and income m. Define it as ${displaystyle mu (p;q,m)=e(p,v(q,m)),!}$ ### Equivalent Variation ${displaystyle EV=mu (p^{0};p^{1},m^{1})-mu (p^{0};p^{0},m^{0}),!}$ note 1 indicates after, 0 indicates before Current prices are the base, what income change will give equivalent utility ### Compensating Variation ${displaystyle CV=mu (p^{1};p^{1},m^{1})-mu (p^{1};p^{0},m^{0}),!}$ New prices are the base, what income change will compensate for price change ### Consumer’s Surplus Utility and Consumer’s Surplus ${displaystyle Delta CS=int limits _{p^{0}}^{p^{1}}{x(t)dt},!}$ Generally ${displaystyle EVgeq CSgeq CV,!}$ When utility is quasilinear ( ${displaystyle U=U(X1)+X0),!}$ , then: ${displaystyle EV=CS=CV,!}$ ## Arrow’s Impossibility Theorem Arrow’s Impossibility Theorem An illustration of the problem of aggregation of social welfare functions: Three individuals each have well-behaved preferences. However, aggregating the three does not produce a well behaved preference function: • Person A prefers red to blue and blue to green • Person B prefers green to red and red to blue • Person C prefers blue to green and green to red. Aggregating, transitivity is violated. • Two people prefer red to blue • Two people prefer blue to green, and • Two people prefer green to red. What does society want? ## Preferences ### Consumption Bundles Define a consumption set X, e.g. {house, car, computer}, ‘x’, ‘y’, ‘z’, are bundles of goods, such as x{house,car}, y{car, computer}, z{house, computer}. Goods are not consumed for themselves but for their attributes relative to other goods We want to find preferences that order the bundles. Utility is ordinal, so we only care about which is greater, not by how much. ### Conditions There are several Conditions on preferences to produce a continuous (well-behaved) utility function. The function f is continuous at the point a in its domain if: • ${displaystyle {underset {xto a}{mathop {lim } }},fleft(xright),!}$ exists • ${displaystyle {underset {xto a}{mathop {lim } }},fleft(xright)=fleft(aright),!}$ If ‘f’ is not continuous at ‘a’, we say that ‘f’ is discontinuous at ‘a’. ## References ## Further Reading Demand ## Individual Demand Functions Budget Line Pivots: To get the demand curve, change relative price (say P1). The Budget Line Pivots and the equilibrium solution changes. Thus consumer demand depends on price and income. Plot ${displaystyle P_{1}}$ vs. ${displaystyle X_{1}*}$ , this is the conventional demand curve we began with. The demand function is a relationship between the quantity of a good/service that an individual will consume at different prices, holding other prices and income constant. Every point on the demand function is a utility maximizing point. In effect, the demand curve is a translation from utility metric space into dollar metric space. Thus, point ‘e’ in the diagram above is a point on the demand curve. To construct the demand curve simply vary the price of one good holding the price of other goods and income constant. In graphical terms this is represented as in the diagrams below. Note that the equilibrium points in the upper diagram have their counterparts in ‘quantity space’ in the lower diagram. Therefore, this shows that prices or expenditure information provides a measure of people’s preferences and can be used in making assessments with respect to valuation. An Engel curve is also associated with the development of the demand curve from the utility maximizing framework. An Engel curve is the locus of combinations of goods that an individual would consume if they were faced with changes in income holding all prices constant. Pictorially this would mean a parallel shift in the budget constraint either up or down if income rises or falls, respectively. This Engel curve is also known as an income-consumption curve. • normal goods: the Engel curve is upward sloping • inferior goods: Engel curve is downward sloping • perfect substitutes: Engel curve is positively slope with a slope value of ${displaystyle P_{1}}$ • perfect complements: Engel curve is positively sloped with a slope equal to ${displaystyle P_{1}+P_{2}}$ Homothetic Preferences depend only on the ratio of goods in the consumption bundle. This means that homothetic preferences will yield straight line Engel curves which pass through the origin. This has the interpretation that if income goes up by a factor ${displaystyle t}$ , the demand bundle goes up by a factor ${displaystyle t}$ . Log-linear preferences are an example of homothetic preferences but not all homothetic preferences are log-linear. The demand curve is defined as the relationship between price and quantity in which the quantity demanded is the unknown and the price is the exogenously given variable. The relationship is represented as: ${displaystyle Q=Q(P)}$ The inverse demand curve is simply the monotone transformation of the ‘ordinary’ demand curve. The inverse demand curve indicates, for each level of demand for good 1, the price which would have to be charged for the consumer to consume a given amount. The inverse demand curve is represented as: ${displaystyle P=P(Q)}$ ## Aggregate Demand Aggregation of demand of private goods Aggregation of demand of public goods Moving from individual to aggregate demand requires that we sum individual demands in some way. The level of aggregation is determined by the nature of the issue at hand. Demand functions can be defined over socio-economic groups, cities, states and economy wide. There are numerous issues of ‘aggregation’ not least of which is how one handles the diversity of consumer preferences while aggregating. One of the interesting issues is how to aggregate given the nature of the good. This is an issue in transportation since some people consider transportation infrastructure ‘quasi-public’ goods. Private goods: if I increase my consumption I reduce the amount available for anyone else, the aggregation from individual to aggregate is to sum horizontally. (Left) This reflects the scarcity of the good. Public goods: if I increase my consumption, the amount available remains, the aggregation from individual to aggregate should be vertical. (Right) P=society’s willingness to pay ## Input Demand Theory Isoquant and Isocost Curves To get the cost curve, change the output level, then the isoquant moves and minimum cost is achieved at different K-L combinations. To date we have looked at demand for consumers. Demand for firms applies similar ideas. For instance, a firm may need to choose a trucking company to ship its goods. It can either approach the problem as cost minimization or profit maximization, which are called Duals of each other, and when solved will produce the same answer. Cost minimization: given the output level Q’, minimize costs. An example of this constrained optimization problem just illustrated is: {displaystyle {begin{aligned}&{text{Min C }}={text{ }}wL+rK\&{text{s}}{text{.t}}{text{. }}F(K,L)=Q’\end{aligned}}} where • K = Kapital • L = Labor • w = wage rate • r = interest rate The Marginal Rate of Technical Substitution (MRTS) = w/r. In a competitive market, and a whole set of associated assumptions, firms maximize profits by producing when Marginal Cost = Marginal Revenue. Profit ${displaystyle Pi =PQ-C(Q)}$ . ## Elasticity The utility function is a representation of consumer preferences and a demand function is the mapping of utility (and hence preferences) into quantity space. The elasticity is a summary measure of the demand curve and it is therefore influenced to a great extent by the underlying preference structure. Elasticity is defined as a proportionate change in one variable over the proportionate change in another variable. It, therefore, provides a measure of how sensitive one variable is to changes in some other variable. For example: • How sensitive are people to purchasing transit tickets if the fare went up 5%, 10% or 50%? • How would the demand for housing change if mortgage rates fell by 30% • How would the demand for international air travel change if airfares went up 15%? All of these questions are really asking, “what is the elasticity of demand with respect to some variable”? Price elasticity of demand (PED) can be defined as[1][2][3] ${displaystyle E_{d}={frac {% {mbox{change in quantity demanded}}}{% {mbox{change in price}}}}={frac {Delta Q_{d}/Q_{d}}{Delta P/P}}}$ ### Own-price Elasticity the price elasticity of demand (own price elasticity) is defined as: ${displaystyle varepsilon _{ii}={frac {p_{i}Delta q_{i}}{q_{i}Delta p_{i}}}={frac {p_{i}partial q_{i}}{q_{i}partial p_{i}}}}$ Note that dq/dp is the slope of the demand function so unless there is a very particular type of demand function the slope is not the same as the elasticity. In general, own price elasticity is negative. An increase in ${displaystyle P_{i}}$ should increase the consumption of ${displaystyle Q_{i}}$ , (all else equal). However it is often referred to as positive, this is just confusing. All goods have a price elasticity, however, if the elasticity is less than -1, than the good is called elastic and if the elasticity is between 0 and -1, then the good is inelastic. This is important when looking at the effect of fuel prices on travel demand. ### Cross-price Elasticity Cross price elasticity examines how the quantity of good i consumed changes as the price of j changes: ${displaystyle varepsilon _{ij}={frac {p_{j}Delta q_{i}}{q_{i}Delta p_{j}}}={frac {p_{j}partial q_{i}}{q_{i}partial p_{j}}}}$ If ${displaystyle P_{j}}$ increases and ${displaystyle Q_{i}}$ increases, then ${displaystyle Q_{i}}$ and ${displaystyle Q_{j}}$ are substitutes. If ${displaystyle P_{j}}$ decreases and ${displaystyle Q_{i}}$ increases, then ${displaystyle Q_{i}}$ and ${displaystyle Q_{j}}$ are complements This is important in examining modal competition. ### Income Elasticity ${displaystyle varepsilon _{iY}={frac {Ypartial q_{i}}{q_{i}partial Y}}}$ If Y increases and Qi increases, then Qi is a normal good. If Y increases and Qi decreases then Qi is an inferior good. Examples are auto ownership, and the difference between new and used cars. 1. Parkin; Powell; Matthews (2002). pp.74-5. 2. Gillespie, Andrew (2007). p.43. 3. Gwartney, James D.; Stroup, Richard L.; Sobel, Russell S. (2008). p.425. ## References • Parkin, Michael; Powell, Melanie; Matthews, Kent (2002). Economics. Harlow: Addison-Wesley. ISBN 0-273-65813-1. Remember, an externality is a cost or benefit incurred by a party due to the decision or purchase of another, who neither obtains the consent of the said party, nor effectively considers the costs and/or benefits to the said party in the decision. ## Positive and Negative Feedback: A Systems Approach ### Equilibrium in a Negative Feedback System Illustration of equilibrium between supply and demand Supply and Demand comprise the economist’s view of transportation systems. They are equilibrium systems. What does that mean? Transportation costs both time and money. These costs are represented by a supply curve, which rises with the amount of travel demanded. As described above, demand (e.g. the number of vehicles which want to use the facility) depends on the price, the lower the price, the higher the demand. These two curves intersect at an equilibrium point. In the example figure, they intersect at a toll of0.50 per km, and flow of 3000 vehicles per hour. Time is usually converted to money (using a Value of Time), to simplify the analysis.

Costs may be variable and include users’ time, out-of-pockets costs (paid on a per trip or per distance basis) like tolls, gasolines, and fares, or fixed like insurance or buying an automobile, which are only borne once in a while and are largely independent of the cost of an individual trip.

It means the system is subject to a negative feedback process:

An increase in A begets a decrease in B. An increase B begets an increase in A.

### Disequilibrium

Positive feedback loop (virtuous circle)

Positive feedback loop (vicious circle)

However, many elements of the transportation system do not necessarily generate an equilibrium. Take the case where an increase in A begets an increase in B. An increase in B begets an increase in A. An example where A an increase in Traffic Demand generates more Gas Tax Revenue (B) more Gas Tax Revenue generates more Road Building, which in turn increases traffic demand. (This example assumes the gas tax generates more demand from the resultant road building than costs in sensitivity of demand to the price, i.e. the investment is worthwhile). This is dubbed a positive feedback system, and in some contexts a “Virtuous Circle”, where the “virtue” is a value judgment that depends on your perspective.

Similarly, one might have a “Vicious Circle” where a decrease in A begets a decrease in B and a decrease in B begets a decrease in A. A classic example of this is where (A) is Transit Service and (B) is Transit Demand. Again “vicious” is a value judgment. Less service results in fewer transit riders, fewer transit riders cannot make as a great a claim on transportation resources, leading to more service cutbacks.

These systems of course interact: more road building may attract transit riders to cars, while those additional drivers pay gas taxes and generate more roads.

One might ask whether positive feedback systems converge or diverge. The answer is “it depends on the system”, and in particular where or when in the system you observe. There might be some point where no matter how many additional roads you built, there would be no more traffic demand, as everyone already consumes as much travel as they want to. We have yet to reach that point for roads, but on the other hand, we have for lots of goods. If you live in most parts of the United States, the price of water at your house probably does not affect how much you drink, and a lower price for tap water would not increase your rate of ingestion. You might use substitutes if their prices were lower (or tap water were costlier), e.g. bottled water. Price might affect other behaviors such as lawn watering and car washing though.

### Examples of Feedback Systems

We explore a few examples related to urban growth, accessibility, electric vehicle adoption, and urban transit.

Traffic congestion and Travel Demand ()
An example of a negative feedback system is Traffic Congestion and Traffic Demand. More congestion limits demand, but more demand creates more congestion.

Bus Bunching (+)
An example of a positive feedback system is Bus bunching. Buses operating with high frequency tend to bunch. A lead bus picks up passengers, each passenger makes the bus slower. As the bus approaches the next stop, there are more passengers waiting. The following bus in contrast gets faster as there are fewer and fewer passengers waiting at each stop, since the previous bus is later and gets more boardings. Eventually the buses bunch together. This simulation illustrates and explains the phenomenon.

Agglomeration Economies and City Growth (+)
Economists have sought to understand why cities grow and why large cities seem to be at an advantage relative to others. One explanation that has received much attention emphasizes the role of agglomeration economies in facilitating and sustaining city growth. The clustering of firms and workers in cities generates positive externalities by allowing for labor market pooling, input sharing, and knowledge spillovers. (Rosenthal & Strange 2004)

Accessibility and Land Value (+)

Accessibility and land value comprise a positive feedback system. Where land is expensive, it is developed more intensively. Where it is more intensively developed, there are more activities and destinations that can be reached in a given time. Where there are more activities, accessibility is higher. Where accessibility is higher, land is more expensive.

Gas tax and electric vehicle usage

Gas Tax and Electric Vehicle Usage (+)
Imagine all gasoline vehicle users pay for all transportation costs. Imagine total expenses are $100,000,000 and the total number of users are 1,000,000, traveling 10,000 miles per year (for a total travel of 10,000,000,000 miles per year, at a cost of$0.01/mile), and all gasoline powered cars get 30 MPG. In that case, if all vehicles are gasoline powered, the gas tax will be $0.30/gallon (or$0.01/mile), in line with current costs. Now imagine, only half of all cars pay the gas tax, the tax jumps to $0.60 to cover costs, still quite tolerable, but as the gas tax rises, the number of gasoline powered cars should be expected to fall. The following image shows the expected gas tax based on the above assumptions with a varying number of gasoline powered cars on the road. Note especially this is a log-log scale. At 50,000 cars with gasoline engines (95% non-gasoline powered), the tax jumps to$6.00 per gallon (above European levels), but the last car has to pay \$300,000 per gallon. The move away from the gas tax is a positive feedback system that will accelerate. A replacement may be required.

Urban Transit (The “Mohring Effect”) (+)
Fixed-route urban transit networks also have the potential to exhibit characteristics of a positive feedback system through a process known as the “Mohring effect”. Since the costs facing users of public transit services include elements such as the user’s time, which has a fixed component to account for time spent waiting for a bus or train, it can be shown that at higher levels of demand waiting times decrease due to more frequent service. This, in turn, makes public transit more attractive. Thus, the setting of marginal cost fares should take into account relevant user time costs in order to achieve optimality. The Mohring effect may provide an efficiency rationale for the subsidization of some urban transit services.

The Mohring effect may arise through the provision of more frequent service on a single route, as described above, and also in a network setting. For example, if the density of routes in a network are increased, the time costs faced by users in accessing the nearest stop will decrease, again reducing the total cost of a trip.

Mode Choice and the Mogridge hypothesis (+)
(see Mogridge, M.J.H., D.J. Holden, J. Bird and G.C. Terzis, 1987, The Downs/Thomson Paradox and the Transportation Planning Process, International Journal of Transport Economics, 14(3): 283-311.)

Why did the automobile take-off? Because at all values of auto-mode share, the automobile has a faster travel speed than transit, even though the city might be better off as a whole (have an overall faster speed) if the congestion from autos was avoided altogether and everyone rode the bus. Mogridge[1] and others have hypothesized that this is indeed how congested transportation systems work in the absence of charges for road access. [1]

Is the left figure correct? Or does the second figure (right) more accurately reflect the empirical evidence? The best collection of evidence to date has been compiled in a review by Mogridge[2].

## Network Externalities

The idea underlying network externalities is that a network is more valuable the more people (destinations) who are on (served by) it.

### Examples of networks

Examples of networks from communications include:

• telegraph,
• telephone,
• fax,
• email,
• World Wide Web,
• automated teller machines, and
• the English language.

In transportation, networks examples include:

• highways,
• airports,
• shipping containers.

### Examples of network externalities

Non-transportation examples of network externalities include

• the typewriter keyboard,
• electrical sockets,
• nuts and bolts,
• weights and measures (SI or the metric system)

and anything else that has been standardized.

Exercise
Identify four technologies (related in some way to transportation) in which network externalities exist (that have inter-organizational standards). […]

### Terms

Terms that are often used in describing network externalities:

## Agglomeration Economies

The idea of agglomeration economies has been long considered in urban economics. Transportation of one form or another drives these agglomeration economies.

The spatial economics literature observes: Specialized and diversified cities co-exist. Larger cities tend to be more diversified. The distribution of city-sizes and specializations tend to be stable over time. City growth is related to specialization and diversity. Relocations are from diversified to specialized cities. Assumptions include: crowding, agents, labour mobility, (endogenous) self-organisation, path-dependency, systems of cities (policentricity).

The following are some quotes about agglomeration:

• “My purpose is to show that cities are primary economic organs” (Jacobs 1969, p.6).
• “Development is a process of continuously improving in a context that makes injecting improvisations feasible. Cities create that context. Nothing else does” (Jacobs 1984, p.155).
• “The city is not only the place where growth occurs, but also the engine of growth itself” (Duranton 2000, p.291-292).
• “Large cities have been and will continue to be an important source of economic growth” (Quigly 1998, p.137).
• “Agglomeration can be considered the territorial counterpart of economic growth” (Fujita and Thisse 2002, p.389).

Agglomeration, productivity and (urban) scale in a knowledge driven economy

• “City-regions are locomotives of the national economies within which they are situated, in that they are the sites of dense masses of interrelated economic activities that also typically have high levels of productivity by reason of their jointly-generated agglomeration economies and their innovative potentials ” Scott and Storper, 2003
• “Metropolitan spaces are becoming, more and more, the adequate ecosystems of advanced technology and economy…. [T]he decrease of communication costs does not by itself lead to a spreading and diffusion of wealth and power; on the contrary, it entails their polarization.” Veltz, 2005

The following table enumerates different types of scale (intra-firm) and agglomeration (inter-firm) scale economies.

 ‘ Type of scale economy Example Internal 1. Pecuniary Being able to purchase intermediate inputs at volume discounts Technological 2. Static technological Falling average costs because of fixed costs of operating a plant 3. Dynamic technological Learning to operate a plant more efficiently over time External or Agglomeration Localization Static 4. Shopping Shoppers are attracted to places where there are many sellers 5. Adam Smith Specialization Outsourcing allows both the upstream input suppliers and downstream firms to profit from productivity gains because of specialization 6. Marshall labor pooling Workers with industry-specific skills are attracted to a location where there is a greater concentration.a Dynamic 7. Marshall-Arrow-Romer Learning-by-doing Reductions in costs that arise from repeated and continuous production activity over time and which spill over between firms in the same place Urbanization Static 8. Jane Jacobs innovation The more that different things are done locally, the more opportunity there is for observing and adapting ideas from others 9. Marshall labor pooling Workers in an industry bring innovations to firms in other industries; similar to no. 6 above, but the benefit arises from the diversity of industries in one location. 10. Adam Smith division of labor Similar to no. 5 above, the main difference being that the division of labor is made possible by the existence of many different buying industries in the same place Dynamic 11. Romer endogenous growth. The larger the market, the higher the profit; the more attractive the location to firms, the more jobs there are; the more labor pools there, the larger the market—and so on 12. Pure agglomeration Spreading fixed costs of infrastructure over more taxpayers; diseconomies arise from congestion and pollution

Source: World development report 2009: reshaping economic geography By World Bank [3], Adapted from Kilkenny 1998[4]

## Standardization and Coordination Externalities

Image of Calculator Keypad: 0, 1, 2, 3 increasing from bottom to top.

Image of Telephone Keypad: 1, 2, 3 increasing from top to bottom, with 0 at bottom.

A typical remote control for Cable TV in the first part of the 21st century has up and down arrows to adjust channels. Pushing the up (plus) button will move you away from channel 0, while the down button will move you toward channel 0 (although if you reach the final channel, you will return to home). But remote controls also have a navigation for the onscreen guide – these have an up, down, left, and right arrow. The up arrow moves you through the onscreen guide, but here up move you toward channel 0, while down moves you away from 0. The left and right arrows move you forward or backward through time.

These remote controls have a further set of controls to operate an auxiliary device like a DVD or an inbuilt device like a personal video recorder. The left arrow, following the convention from tape recorders, plays (forward in time), while the double left arrow (on the right-most side) is fast forward and the double right-pointing arrow (on the left side) moves you in reverse (rewind). Other buttons do other things.

Complaints about the complexity of modern remote controls are hardly unique[5] . Each remote is custom for a particular box, so as people accumulate boxes attached to TVs, the number of remotes increases accordingly. The utopia of the universal remote remains unreached; one hopes the situation will not sustain for another few decades before standardization moves in, or some other interface becomes widespread.

Like remote controls, keypads are another area where conventions may confuse.

Keypads on telephones and calculators represent the same ten digits, however they have different layouts. The telephone keypad, introduced with the advent of Touch Tone dialing by the Bell System in the 1960s places 0 (or O for operator, it is not always clear on telephones) at the bottom, and then numbers digits 1 – 9 in three rows of three columns each from the top. A calculator keypad (also used on computer keyboards) on the other hand, while it places 0 at the bottom, numbers 1 to 9 also in three rows of three columns, but in this case beginning at the bottom, as shown in Figure. These conventions have carried over to computers, which could array numbers in any random way, but use the different conventions to represent the different devices. Newer devices, such as television remote controls, could use either, but typically follow the telephone layout (though some have original layouts themselves, e.g. going from 1 to 4 on the first row, 5 to 8 on the second row, and 9 and 0 on the third row).

For operating a television, rarely an urgent activity, the additional cognitive load of a poorly-designed or non-standard interface is annoying, but not dangerous. With the case of election ballots, such confusion and resulting error may change the outcomes (such as the odd butterfly ballot used in West Palm Beach, Florida in the 2000 Presidential election, resulting in a disproportionate (compared to other jurisdictions) number of votes for Pat Buchanan, and likely giving the state of Florida, and thus the United States electoral college and the presidency to George W. Bush).

American travelers trying to write emails in some European countries may note that the standard QWERTY keyboard found in the English speaking world (so-named for the keys on the top-row of letters) has been replaced by a keyboard, which mainly swaps the Y and Z, but has some minor changes, dubbed the QWERTZ kezboard. This is just enough to throw off touch-tzpists (er, typists). I am sure the confusion is two-way.

For driving cars in the United States, many functions have been fortunately standardized. The brake foot pedal is on the left, the accelerator on the right. The steering wheel itself usually performs as expected. Less critical functions remain confusing, especially when switching cars, or driving an unfamiliar vehicle, such as a rental car, the difficulty compounds as this is usually done in an unfamiliar place. Where is the windshield wiper? The light switch? The brights? The transmission control? The radio? The environmental controls? The locks? The window controls? The rear-view window control? The unlock for the trunk? The unlock for the gas tank? Where is the gas tank – driver or passenger side? All vary with make, model, and year of vehicle.

Driving on the left of the right is standardized locally, but not globally. As any traveler from continental Europe, North America, or South America knows, things differ on the islands of Great Britain, New Zealand, Japan, the Caribbean, and even the island-continent of Australia and the Indian subcontinent.

Traffic signals usually report red on top and green on bottom. What does it mean when the light is simultaneously red and green? Or red and yellow (amber), or green and yellow? Or the green light flashes? All of these patterns are local, but not global standards.

## Construction of Revealed Demand (Fulfilled Expectation) Curve with Positive Network Externalities

Construction of Revealed Demand (Fulfilled Expectation) Curve with Positive Network Externalities

(based on Economides, Nicholas (1996) The Economics of Networks. Journal of Industrial Organization, Vol. 14, no. 6, pp. 673-699 October 1996)

A demand curve for a typical good is downward sloping, the more it costs, the less that will be consumed. However, the demand for a network good rises with the number of members of the network (Economides 1996). Each user of the network creates a positive externality for other users. Thus, networks exhibit a seemingly upward sloping demand curve, self-limiting at saturation, with perfectly inelastic demand.

### Rationale

Figure 1 constructs the revealed demand curves for positive network externalities. Let

${displaystyle P(n;n_{e})}$

be the willingness to pay for the nth unit of the good when ne units are expected to be sold (assume each consumer purchases only one unit of the good). The network is more valuable the more units are sold. With only one consumer, (

${displaystyle n=1}$

), the network is not particularly valuable, so the implicit demand at

${displaystyle n=1}$

(

${displaystyle D_{1}}$

) is low, lower than at

${displaystyle D_{2}}$

, which is lower than

${displaystyle D_{3}}$

, etc. Drawing a line between the number of consumers (

${displaystyle n}$

) and the implicit demand curve at that number (

${displaystyle D_{n}}$

) traces out an approximately parabolic shape,

${displaystyle P(n,n)}$

.

### Conditions

${displaystyle P(n,n)}$

is the equilibrium price where the demand curve for a network of size

${displaystyle n}$

(

${displaystyle D_{e}}$

) intersects the vertical projection of the network size when the number of consumers (network size) is

${displaystyle e}$

.

${displaystyle P(n,n)}$

is thus the fulfilled expectations (or revealed demand) curve, the set of prices that the nth consumer would actually pay to join the network which would sustain n-consumers. The fulfilled expectations demand is increasing for small

${displaystyle n}$

if any one of three conditions hold:

1. “The utility of every consumer in a network of zero size is zero, or
2. there are immediate and large external benefits to network expansion for very small networks, or
3. there is a significant density of high-willingness-to-pay consumers who are just indifferent on joining a network of approximately zero size.”

### Saturation

While demand rises with the number of members, thereby exhibiting positive critical mass under perfect competition, there is a saturation point, such that increasing the number of members does not add value. Such a system exhibits multiple equilibria (the largest of which is stable), and under perfect competition, the amount of network may be under-supplied because the positive externalities cannot be internalized to the producing firms.

### Intersection with U-shaped cost curves

A – supply intersects demand twice: when cost is decreasing and demand is increasing, and when cost is increasing and demand is decreasing

B – supply intersects demand twice: when cost is decreasing and demand is decreasing, and when cost is increasing and demand is decreasing

C – supply intersects demand twice: when cost is decreasing and demand is increasing, and when cost is increasing and demand is increasing

D – supply does not intersect demand

We might then think about intersecting our parabolic demand curve with our U-shaped supply curve. Ignoring tangencies, four key outcomes are possible, as shown in the figures below. In the three cases (A,B,C) where the curves intersect, the intersection on the right side, denoted Q*, would be a stable equilibrium. However, to get to the intersection on the right, one might have to pass through the intersection on the left.

### Analogy between Scale and Scope economies on the cost side

In the chapter on costs we noted that there exist scale and scope economies on the cost side. Scale economies indicate it is cheaper to produce a given amount if more units are being produced (as a fixed cost can be spread over more units) and scope economies indicate it is cheaper to produce multiple goods together rather than separately.

On the demand side, we noted above network externalities, which are analogous to scale economies, it is more valuable to consume the more consumers there are. Goods may also be more valued if consumed together rather than separately (e.g. complements) or because variety is preferred to monotony. These Variety or Inter-technology externalities are analogous to economies of scope.

## Other Concepts Related to Positive Feedback Systems

### Companion-Innovation

(based on Garrison W, Souleyrette R, (1996), “Transportation, Innovation and Development: The Companion Innovation Hypothesis”, Logistics and Transportation Review, vol. 32, pp. 5-38).

The economy is a series of linked markets. The “companion innovation” hypothesis suggests that improvements in transport energize other sectors of the economy.

Does the demand curve include those positive externalities?

How smart are markets?

Does willingness to pay change over time at a rate greater than the discount rate?

The reorganization and innovation lead to productivity growth, which should be captured from a macro-economic perspective (see the lecture on transportation and productivity). If positive externalities are not captured (and negative are), there is clearly underinvestment.

### Learning Curves

Average variable costs decline with output and time as processes get more efficient (people get smarter). Research and Development is a function of market size, which helps explain the process.

### Consumption Economies

Average fixed costs decline with market size.

In markets with large fixed costs that have cost recovery as an aim (public infrastructure as an example), this can be very important. As the market grows, the cost per user drops. This of itself should increase demand – and can be seen as a positive consumption scale externality

### Increasing and Decreasing Returns and Equilibrium

(see Arthur, Brian (1990) Increasing Returns and Path Dependence in the Economy. The University of Michigan Press.)

## How Networks Grow

To start, a network must have value to some network members at a minimal size (exceeding the cost of joining), or it must be subsidized. Success conditions for a new network suggest

1. it must either be compatible with existing networks (i.e. not really so new), or
2. be significantly more valuable to get people to adopt it.

For instance, the interstate highways were compatible with the existing vehicle highway system, interchanges were built, and the same cars could use both. Railroads on the other hand were very valuable compared with canals and animal led carts against which they were initially competing, enabling their success despite the incompatibility of the technologies. In short, if compatibility has costs, it can limit the market because of the extra handling costs, additional waiting time, or an additional layer of processing (such as software) required to decode things.

Thomas Hughes said “Mature systems suffocate nascent ones”.[6] This means that well-developed technologies occupying a particular niche make it very hard for a new technology to move into that niche, since it does not have all of the compatible infrastructure and correlated compatible technologies.

### Where Does Intelligence Lie

• Smart Networks, Dumb Packets/Vehicles (Railroads, Telephone)
• Smart Packets/Vehicles, Dumb Networks (Roads, Internet)

Important to resolve this in network design

### Network Design vs. Network Growth

Network Design Problem (NDP) tries to determine “optimal” network according to some criteria (Z). – Normative

E.g. Maximize Z, subject to some constraints.

Network Growth Problem tries to predict actual network according to observed or hypothesized behaviors. – Positive

### Questions

• Why do networks expand and contract?
• Do networks self-organize into hierarchies?
• Are roads an emergent property?
• Can investment rules predict location of network expansions and contractions?
• How can this improved knowledge help in planning transportation networks?
• To what extent do changes in travel demand, population, income and demographic drive changes in supply?
• Can we model and predict the spatially specific decisions on infrastructure improvements?

### Network Growth

• Depends on existing and forecast transportation demand
• Depends on existing transportation supply
• Network can be viewed as output of a production function: N = f( D, S)

Over the long term, we expect networks to grow in the fashion of an S-Curve as discussed in the Lesson on Positive Externalities

### How networks change with time

• Nodes: Added, Deleted, Expanded, Contracted
• Flows: Increase, Decrease

### The Node Formation Problem

• Christaller’s Central Place Theory (CPT) sought to answer: How are urban settlements spaced, more specifically, what rules determine the size, number and distribution of towns? Christaller’s model made a number of idealizing assumptions, especially regarding the ubiquity of transport services, in essence, assuming the network problem away. His world was a largely undifferentiated plain (purchasing power was spread equally in all directions), with central places (market towns) that served local needs. The plain was demarcated with a series of hexagons (which approximated circles without gaps or overlaps), the center of which would be a central place. However some central places were more important than others because those central places had more activities. Some activities (goods and services) would be located nearer consumers, and have small market areas (for example a convenience store) others would have larger market areas to achieve economies of scale (such as warehouses).

### Central Place & Network Hierarchy

Network Hierarchy is much like Central Places (Downtown Minneapolis, Suburban Activity Centers (e.g. Bloomington, Edina, Eden Prarie), Local Activity Centers (e.g. Dinkytown, Stadium Village, Midway), Neighborhood Centers (4th Avenue & 8th Street SE).

Central Places occur both within and between cities.
Hierarchy: Minneapolis-St. Paul; Duluth, St. Cloud, Rochester; Morris, Brainerd, Marshall, etc.; International Falls, etc.

## Thought Question: Applications of Positive Externalities

• Do Positive Externalities Exist (or are they Internalized?) Discuss …
• What does this say for the prospects of Intelligent Transportation Systems?
• What are the prospects for Automated Highway Systems as opposed to Intelligent Vehicles (and relatively Dumb Roads)?

## References

1. ab Mogridge, Martin J.H.; Holden, D.J.; Bird, J.; Terzis, G.C. (October 1987). “The Downs/Thomson paradox and the transportation planning process”. International Journal of Transport Economics 14 (3): 283-311.
2. Mogridge, Martin J.H. (January 1997). “The self-defeating nature of urban road capacity policy: a review of theories, disputes and available evidence”. Transport Policy 4 (1): 5-23. doi:10.1016/S0967-070X(96)00030-3.
4. Kilkenny, Maureen (1998) “Economies of Scale” Lecture for Economics 376, Rural, Urban, and Regional Economics, Iowa State University, Ames Iowa
5. Nielson, J. (2004), ‘Remote Control Anarchy’.
6. Hughes, Thomas Parke (2004) American genesis: a century of invention and technological enthusiasm 1870-1970 p. 461

Other Resources in Transportation Economics