# Transportation Economics/Production

Production

## Introduction

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:
* by polluting, systems consume health and increase morbidity and mortality;
* by being dangerous, they consume safety and produce injuries and fatalities;
* by being loud they consume quiet and produce noise (decreasing quality of life and property values); and
* by emitting carbon and other pollutants, they harm the environment.
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 and Production

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.
T = g(K, L, E, M,)
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.
Measuring inputs and outputs
* material inputs -- volume/mass
* human inputs--labor and users (time)
* service inputs - navigation, terminal operations
* capital inputs - physical units, monetary units (stocks & flows)
* design inputs - dimensions, weight, power
* transportation - cargo trips, vehicle trips, vehicle miles, capacity miles, miles

## Characterizing Transportation Production

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).
Inputs are supplied by carriers, users, and public:
* carriers: terminal activities, line haul activities, etc.
* users: the value of time, etc.
* public: infrastructure
Production is characterized by multidimensional (heterogeneous) outputs.
* quantity: most common measures of outputs;
** tonne-kilometres
** passenger-kilometres
* spatial dimension - origin-destination and direction
* time dimension - transit time, peaking and seasonality
* quality of service - speed, reliability, etc.
Examples of the use of the production approach for system design considering both inputs and outputs are illustrated in the following table:
| Inputs | Outputs |
| dimensions | surface area/volume carrying capacity |
| size, speed | transport capacity (e.g. vehicles per hour) |
| system capacity, infrastructure quality | traffic flow |
| capacity, vehicle movements | O-D trips |
| runways, terminals | passenger and aircraft movements |
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:
* lumps of capacity and nonstorability of output (mismatch between demand and production quantity)
* joint production of backhaul capacity
* common production; e.g., short haul markets served in conjunction with a longer haul market.

## Production Theory

Theory of production analyzes how a firm, given the given technology, transform its inputs ($x$) into outputs ($y$) in an economically efficient manner. A production function, $y=f(x)$, is used to describe the relationship between outputs and inputs.

## Efficiency

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.

## Production Possibilities Set

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.
linear homogeneous in input prices
$C\left(Q,2P_{1},2P_{2}\right)=2C\left(Q,P_{1},P_{2}\right)$
marginal cost is positive for all outputs
${\frac {\partial C}{\partial Q_{j}}}>0\forall j$.
The derivative of the cost function with respect to the price of an input yields the input demand function.
${\frac {\partial C}{\partial P_{j}}}=X\left(\bullet \right)$
As input prices rise we always substitute away from the relatively more expensive input.
${\frac {\partial ^{2}C}{\partial P_{i}^{2}}}\leq 0\forall i$

## Functional Forms

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.

### Linear

A linear production function is the simplest:
    $Y=A+\alpha L+\beta K$

### Quadratic

A quadratic production function adds squares and interaction terms.
    $Y=A+\alpha L+\beta K+\gamma L^{2}+\phi K^{2}+\rho KL$

### Cobb-Douglas

(adapted from Wikipedia article on the Cobb Douglas function )
For production, the Cobb-Douglas function is
    $Y=AL^{\alpha }K^{\beta },$
where:
* Y = total production (the monetary value of all goods produced in a year)
* L = labor input
* K = capital input
* A = total factor productivity
* $\alpha$ and $\beta$ are the output elasticities of labor and capital, respectively. These values are constants determined by available technology.
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 $\alpha =0.15$, a 1% increase in labor would lead to approximately a 0.15% increase in output.
Further, if:
    $\alpha +\beta =1$
the production function has constant returns to scale. That is, if L and K are each increased by 20%, Y increases by 20%. If
    $\alpha +\beta <1$
returns to scale are decreasing, and if
    $\alpha +\beta >1$
returns to scale are increasing. Assuming perfect competition and$\alpha +\beta =1$ , $\alpha$ and $\beta$ can be shown to be labor and capital's share of output.

### Translog

The translog production function is a generalization of the Cobb–Douglas production function. The name translog stands for 'transcendental logarithmic'.
The two-factor translog production function is:
    ${\begin{aligned}\ln(Y)&=\ln(A)+a_{L}\ln(L)+a_{K}\ln(K)+b_{LL}\ln(L)\ln(L)\\&\qquad+b_{KK}\ln(K)\ln(K)+b_{LK}\ln(L)\ln(K)\\&=f(L,K).\end{aligned}}$
where L = labor, K = capital, and Y = product.

### CES (constant elasticity of substitution)

Constant elasticity of substitution (CES) function: $Y=A[\alpha K^{\gamma }+(1-\alpha )L^{\gamma }]^{\frac {1}{\gamma }}$
$\gamma =0$ corresponds to a Cobb–Douglas function, $Y=AK^{\alpha }L^{1-\alpha }$

### Leontief

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
    $Y=\min(aK,\ \ bL,\ \ ...).$

## Characteristics of a Production Function

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)
$Q=f(K,L)$
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 $\Delta Q/\Delta L$ which is defined as the marginal product of labor and the $Q/L$ 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, $Q/L$ 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 $MP=0$, $AP>0$ yet TP is decreasing.
The principle of "diminishing marginal productivity " is well illustrated here. This principle states that as you add units of a variable factor to a fixed factor initially output will rise, and most likely at an increasing rate but not necessarily) but at some point adding more of the variable input will contribute less and less to total output and may eventually cause total output to decline (again not necessarily).
Any shifts in the fixed factor (or technology) will result in an upward shift in TP, AP and MP functions. This raises the interesting and important issue of what it is that generates output changes; changes in variable factors, technology and/or changes in technology.

## Isoquants

The isoquant reveals a great deal about technology and substitutability. Like indifference curves, the curvature of the isoquants indicate the degree of substitutability between two factors. The more 'right-angled' they are the less substitution. Furthermore, diminishing marginal product plays a role in the slope of the isoquant since as the proportions of a factor change the relative Marginal Product's change. Therefore, substitutability is simply not a matter of the technology of production but also the relative proportions of the inputs.
Rather than consider one factor variable, consider two (or all) factors variable.
$Q=f(K,L)$.
Taking the total derivative and setting equal to zero
$dQ={\frac {\partial f}{\partial K}}dK+{\frac {\partial f}{\partial L}}dL=0$
rearranging one can see that the ratio of the marginal productivities (${\frac {MP_{K}}{MP_{L}}}$) equals ${\frac {dk}{dL}}$
Equivalently, the isoquant is the locus of combinations of K and L which will yield the same level of output and the slope (${\frac {dk}{dL}}$) 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.

## Constrained Optimization

An example of this constrained optimization problem just illustrated is:
${\begin{aligned}&{\text{Min cost }}={\text{ }}p_{1}x_{1}+p_{2}x_{2}\\&{\text{s}}{\text{.t}}{\text{. }}F(x_{1},x_{2})=Q\\\end{aligned}}$
where
* f is the production function
* Objective function (Min cost): desire
* Constraint (subject to): necessity
* $x_{1},x_{2}$: decision variables
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.
${\begin{aligned}&{\text{Max }}g\left({\bar {x}}\right)\\&{\text{s}}{\text{.t}}{\text{. }}h_{j}\left({\bar {x}}\right)=b_{j}\\\end{aligned}}$
Lagrangian:
${\text{Max}}\Lambda \left({\bar {x}},{\bar {\lambda }}\right)=g\left({\bar {x}}\right)-\sum {{\bar {\lambda }}_{j}\left(h_{j}\left({\bar {x}}\right)-b_{j}\right)}$
To find the maximum, take the first derivative and set equal to zero
${\frac {\partial \Lambda }{\partial x_{i}}}={\frac {\partial g}{\partial x_{i}}}-\sum \limits _{j}{\lambda _{j}}{\frac {\partial h_{j}}{\partial x_{i}}}=0$
${\frac {\partial \Lambda }{\partial \lambda _{j}}}=-h_{j}\left(x\right)+b_{j}=0$
* Lagrangian is maximized (minimized)
* Lagrangian equals the original objective function
* constraints are satisfied
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.

### Example

${\begin{aligned}&{\text{Min cost }}={\text{ }}p_{1}x_{1}+p_{2}x_{2}\\&{\text{s}}{\text{.t}}{\text{. }}F(x_{1},x_{2})=Q\\\end{aligned}}$
$\Lambda =p_{1}x_{1}+p_{2}x_{2}-\lambda (F(x_{1},x_{2})-Q)$
${\frac {\partial \Lambda }{\partial x_{1}}}=-p_{\text{1}}-\lambda {\frac {\partial F\left(x_{\text{1}},x_{\text{2}}\right)}{\partial x_{\text{1}}}}=0$
${\frac {\partial \Lambda }{\partial x_{2}}}=-p_{\text{2}}-\lambda {\frac {\partial F\left(x_{\text{1}},x_{\text{2}}\right)}{\partial x_{\text{2}}}}=0$
${\frac {\partial \Lambda }{\partial \lambda _{j}}}=Q-F\left(x_{\text{1}},x_{\text{2}}\right)=0$
${\frac {p_{\text{1}}}{\frac {\partial F\left(x_{\text{1}},x_{\text{2}}\right)}{\partial x_{\text{1}}}}}=\lambda$
${\frac {p_{2}}{\frac {\partial F\left(x_{\text{1}},x_{\text{2}}\right)}{\partial x_{2}}}}=\lambda$
${\frac {p_{\text{1}}}{p_{2}}}={\frac {\frac {\partial F\left(x_{\text{1}},x_{\text{2}}\right)}{\partial x_{\text{1}}}}{\frac {\partial F\left(x_{\text{1}},x_{\text{2}}\right)}{\partial x_{2}}}}$
so
$\lambda ={\frac {\partial \Lambda }{\partial Q}}$
is equal to the marginal cost of output.

### Conditions

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.

## Optimization

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.
$\Pi =Pf\left(K,L\right)-\left(wL+rK\right)$
${\frac {\partial \Pi }{\partial K}}=P{\frac {\partial f}{\partial K}}-r=0$
${\frac {\partial \Pi }{\partial L}}=P{\frac {\partial f}{\partial L}}-w=0$
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 function
* The budget constraint
* The expansion path
The 'production cost function' is the lowest cost at which it is possible to produce a given output.

## Duality

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.

## Factor Demand Functions

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.

## Input Cost Functions

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

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.
Types of Technical Change:
* secular - include time in production function
* innovation - include presence of innovation in production function
* facility - include availability of facility in production function

### Conditions

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.