01 - Production and cost functions in transport - DME

Production function

production function ($x$)

The production function is the relation between the output (products, $x$ ) and the Capital ( $K$ ) and Labor ( $L$ ) involved in making that product.

$x = f (K, L)$

It works under the assumption of efficient process: given $K$ and $L$ , we have the best output possible.

Indicators

Average Productivity

average productivity

$P_{a v, L} = \frac{x}{L} P_{a v, K} = \frac{x}{K}$

Marginal Productivity

marginal productivity ($p_{ma,l}$,$p_{ma,k}$ )

Marginal productivity is a measure of how many units of input are necessary to increase the output of 1 unit.

$P_{m a, L} = \frac{\partial x}{\partial L} P_{m a, K} = \frac{\partial x}{\partial K}$

Characteristics

Usually, a #Production function tends to have the following characteristics:

Monotony of a production function

monotony of the production function

The #Production function increases when the labor or the capital increases, or, at the very least, it does not increase. It NEVER decreases.
$\frac{\partial x}{\partial L} \geq 0 \frac{\partial x}{\partial K} \geq 0$

Convexity of a production function

convexity of the #production function

The product $X$ grows slower and slower as the Labor or the Capital keep increasing.

$\frac{\partial^{2} x}{\partial^{2} L} = \frac{\partial P_{m a, L}}{\partial L} \leq 0 \frac{\partial^{2} x}{\partial^{2} K} = \frac{\partial P_{m a, K}}{\partial K} \leq 0$

Iso-production curves

They show what are the admissible combinations of labor and capital for a given value of the #Production function.

Diminishing return law

Assuming a production function x=f(K,L), if ΔL (or Δk) and K is constant (or L), then productivity increases proportionally more than the productive factor until at a level upon which the productivity diminishes.

Return to scale

Return to scale is the concept of relating how fast the #Production function increases with the increase of the inputs:

Increasing return to scale

We have an increasing return to scale when the #Production function increases more than proportionally with the inputs.

Constant return to scale

We have a constant return to scale when the #Production function increases proportionally with the inputs.

Reducing return to scale

We have a reducing return to scale when the #Production function increases less than proportionally with the inputs.

Types of function

There are a few #Iso-production curves that are often used in the field:

#Cobb-Douglas
#Leontief

Cobb-Douglas

The Cobb-Douglas #Iso-production curves has the following equation for the [[#Production function]]

f (K, L) = x = K^{a} L^{b}

where:

$a, b :$ are 2 parameters

The Cobb-Douglas function can be written linearly:

\begin{aligned} x & = K^{a} L^{b} = \\ \underset{\hat{x}}{\underset{⏟}{\ln (x)}} & = a \underset{\hat{K}}{\underset{⏟}{\ln (K)}} + b \underset{\hat{L}}{\underset{⏟}{\ln (L)}} = \\ \hat{x} & = a \hat{K} + b \hat{L} \end{aligned}

In the Cobb-Douglas function, the #Return to scale is:

R T S = - \frac{a}{b} \frac{L}{K}

Leontief

The Leontief [[#Production function]] is often used for its semplicity:

f (K, L) = x = min (a L, b K)

Technical Substitution Relation (TSR)

Cost Function

01 - Production and cost functions in transport - DME

cost function ($ct$)

The Cost Function, or Total cost ( $C T$ ) is the relation
$C T (x, w, r) = w L + r K$
where:

$L, K :$ number of productive factors
$r, w :$ unitary cost of the productive factors

We work under the assumption that we have the minimum cost for producing $x$ .

Indicators

Average cost

average cost ($c_{av}$)

$C_{a v} = \frac{C T}{x}$

Marginal cost

marginal cost ($c_{ma}$)

$C_{m a} = \frac{\partial C T}{\partial x}$

Properties of the cost function

???

teorema

There is no diminishing with unitary cost of the productive factors

[?] What does this mean?

Homogeneity of the cost function

homogeneity

The #Cost Function is homogeneous of first grade in $r$ and $w$ .

Convexity of the cost function

convexity

The [[#Cost Function]] is convex in $w$ .

Continuity of the cost function

continuity

The [[#Cost Function]] is continous in $w$ for $w \geq 0$ .

Fixed and variable costs

The total cost can be divided into the following sum:

C T = C F + C V

where

$C F :$ #Fixed cost
$C V :$ #Variable cost

Fixed cost

fixed cost

Fixed costs ( $C F$ ) are costs that do not change in a given system. They can only be defined in the #Short term cost function.

Variable cost

Variable costs ( $C V$ ) are costs dependent on the production of a system.

They exist both in the [[#Short term and long term]] and the [[#Long term cost function]].

Short term and long term

Short term cost function

In the short term, the #Variable cost and the #Fixed cost:

The graph specifically shows the sum of the Average Fixed cost and the average Variable cost to give, in blue, the #Average cost.

Long term cost function

In the long term, even the #Fixed cost can change. Therefore, there are not #Fixed cost in the Long term cost function, only #Variable cost.

The long term cost function can be obtained overlapping all the [[#Short term cost function]] at different sizes of the system:

Each $x_{i}^{*}$ is the optimal production in the short term, at the given system size. All the $x_{i}^{*}$ give the long term cost function. For a business, the optimal size is that that minimizes the long term cost function.

Efficiency

efficiency

Efficiency is the maximum output for a fixed numer of productive factors.

Types of efficiency

We can define efficiency based on different things:

#Technical efficiency
#Efficiency of market assignation
#Social efficiency
#Efficiency of scale
#Structural Efficiency

Technical efficiency

techincal efficiency

Technical efficiency is reached when the cost is minimized.

Efficiency of market assignation

When efficient market decisions are made.

Externalities are included

Efficiency of scale

current capacity vs ideal capacity

Ideal: combination of inputs and outputs inside the constant returns to scale
Long-term competitive equilibrium, where production has constant returns to scale

Structural Efficiency

Structural efficiency is reached when production is located in the non-congested region of possibile inputs combinations.

How to measure efficiency

Economy of scale

#Average cost $↘ ⟺ ↗$ scale

If the nature of a business is to take advantage of economy of scale, that business will try to grow. There will than be just a few big businesses in that sector.

Economy of density

Economy of density is the concept in which a business sees its cost decreasing when the density of the business increases.

The total cost to transport passengers decreases by increasing utilisation of existing vehicle fleet and infrastructure capacity within a market area of given size.

Mohring effect

❗❗❗❗❗❗❗❗❗❗❗❗
❗❗❗ COMPLETARE ❗❗❗
❗❗❗❗❗❗❗❗❗❗❗❗

01 - Production and cost functions in transport - DME

Production function

Indicators

Average Productivity

Marginal Productivity

Characteristics

Monotony of a production function

Convexity of a production function

Iso-production curves

Diminishing return law

Return to scale

Increasing return to scale

Constant return to scale

Reducing return to scale

Types of function

Cobb-Douglas

Leontief

Technical Substitution Relation (TSR)

Cost Function

Indicators

Average cost

Marginal cost

Properties of the cost function

???

Homogeneity of the cost function

Convexity of the cost function

Continuity of the cost function

Fixed and variable costs

Fixed cost

Variable cost

Short term and long term

Short term cost function

Long term cost function

Efficiency

Types of efficiency

Technical efficiency

Efficiency of market assignation

Social efficiency

Efficiency of scale

Structural Efficiency

How to measure efficiency

Economy of scale

Economy of density

Mohring effect