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.

A quadratic production function adds squares and interaction terms.

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 (

) 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[edit]

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[edit]

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

### [edit]

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[edit]

*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[edit]

*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[edit]

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:

provides the optimal plant size.

## Indicators of Aggregate Cost Behavior[edit]

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[edit]

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)[edit]

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)[edit]

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
- if cost elasticity = 1, or
- if cost elasticity > 1, or