Transportation Economics/Production

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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.

[edit] Inputs and Outputs

Goods and bads:

[edit] Inputs

Inputs are goods used in production, bads that are created (eg. pollution)

[edit] Outputs

Outputs are goods that are produced, bads that are eliminated.

[edit] Measurement

Measuring inputs and outputs per unit of time

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

[edit] Aggregation

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.

[edit] Production Possibilities Set

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.

$\frac{\partial ^{2}C}{\partial P_{i}^{2}}\le 0\forall i$

[edit] Functional Forms

Production functions are relationships between inputs and outputs given some technology. A change in technology can effect the production function in two ways. First, it can alter the level of output because it effects 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.

[edit] Cobb-Douglas

A utility function can take on a number of different forms. One of the more popular forms is called the Cobb-Douglas form which is a log-linear function. It is represented as:

$U\left( x_{1}^{{}},x_{2}^{{}} \right)=x_{1}^{c}x_{2}^{d}$

[edit] Translog

[edit] CES (constant elasticity of substitution)

This function describes production, usually at a macroeconomic level, with two inputs which are usually capital and labor. As defined by Arrow, Chenery, Minhas, and Solow, 1961 (p. 230), it is written as

$V = (β * K - ρ + α * L - ρ) - (1 - ρ)$

where

V = value-added, (though y for output is more common),
K is a measure of capital input,
L is a measure of labor input, and
the Greek letters are constants. ^[1]

[edit] Quadratic

[edit] Leontief

$q = min(x 1, x 2)$

[edit] Linear

[edit] Transportation as Input

(see Positive Externalities for more discussion)

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)

[edit] Transportation as Output

Alternatively transportation can be seen as an output, 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,)

[edit] 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, its 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 ∆Q/∆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 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.

[edit] 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.

[edit] Constrained Optimization

An example of this constrained optimization problem just illustrated is:

$\begin{align} & \text{Min cost }=\text{ }p_{1}x_{1}+p_{2}x_{2} \\ & \text{s}\text{.t}\text{. }F(x_{1},x_{2})=Q \\ \end{align}$

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.

$\begin{align} & \text{Max }g\left( {\bar{x}} \right) \\ & \text{s}\text{.t}\text{. }h_{j}\left( {\bar{x}} \right)=b_{j} \\ \end{align}$

Lagrangian:

$\text{Max L}\left( \bar{x},\bar{\lambda } \right)=g\left( {\bar{x}} \right)-\sum{\bar{\lambda }_{j}\left( h_{j}\left( {\bar{x}} \right)-b_{j} \right)}$

To find the maximum, take the first derivative and set equal to zero

$\frac{\partial L}{\partial x_{i}}=\frac{\partial g}{\partial x_{i}}-\sum\limits_{j}{\lambda _{j}}\frac{\partial h_{j}}{\partial x_{i}}=0$

$\frac{\partial L}{\partial \lambda _{j}}=-h_{j}\left( x \right)+b_{j}=0$

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.

[edit] Example

$\begin{align} & \text{Min cost }=\text{ }p_{1}x_{1}+p_{2}x_{2} \\ & \text{s}\text{.t}\text{. }F(x_{1},x_{2})=Q \\ \end{align}$

$L = p 1 x 1 + p 2 x 2 - λ(F (x 1, x 2) - Q)$

$\frac{\partial L}{\partial x_{1}}=-p_{\text{1}}-\lambda \frac{\partial F\left( x_{\text{1}},x_{\text{2}} \right)}{\partial x_{\text{1}}}=0$

$\frac{\partial L}{\partial x_{2}}=-p_{\text{2}}-\lambda \frac{\partial F\left( x_{\text{1}},x_{\text{2}} \right)}{\partial x_{\text{2}}}=0$

$\frac{\partial L}{\partial \lambda _{j}}=Q-F\left( x_{\text{1}},x_{\text{2}} \right)=0$

$\frac{p_{\text{1}}}{\frac{\partial F\left( x_{\text{1}},x_{\text{2}} \right)}{\partial x_{\text{1}}}}=\lambda$

$\frac{p_{2}}{\frac{\partial F\left( x_{\text{1}},x_{\text{2}} \right)}{\partial x_{2}}}=\lambda$

$\frac{p_{\text{1}}}{p_{2}}=\frac{\frac{\partial F\left( x_{\text{1}},x_{\text{2}} \right)}{\partial x_{\text{1}}}}{\frac{\partial F\left( x_{\text{1}},x_{\text{2}} \right)}{\partial x_{2}}}$

so

$\lambda =\frac{\partial L}{\partial Q}$

is equal to the marginal cost of output.

[edit] 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.

[edit] 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:

1. The production function 2. The budget constraint 3. The expansion path

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

[edit] 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.

[edit] 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.

[edit] 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.

[edit] Technical Change

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

Technical change can effect all factors in the production function and thus be 'factor neutral' or it may effect 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 left.

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

[edit] 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.

[edit] References

↑ http://economics.about.com/cs/economicsglossary/g/ces_p.htm

[0] ttp://economics.about.com/cs/economicsglossary/g/ces_p.htm

[1]

Transportation Economics/Production

From Wikibooks, the open-content textbooks collection

Contents

[edit] Inputs and Outputs

[edit] Inputs

[edit] Outputs

[edit] Measurement

[edit] Aggregation

[edit] Production Possibilities Set

[edit] Functional Forms

[edit] Cobb-Douglas

[edit] Translog

[edit] CES (constant elasticity of substitution)

[edit] Quadratic

[edit] Leontief

[edit] Linear

[edit] Transportation as Input

[edit] Transportation as Output

[edit] Characteristics of a Production Function

[edit] Isoquants

[edit] Constrained Optimization

[edit] Example

[edit] Conditions

[edit] Optimization

[edit] Duality

[edit] Factor Demand Functions

[edit] Input Cost Functions

[edit] Technical Change

[edit] Conditions

[edit] References

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