Production Function Definition
- A mathematical representation of the production process.
- It indicates the highest output a firm can produce for a specified combination of inputs, given the state of technology.
Q = f (L,C) where L =Labor and C= capital
Simplified Production Process
- Inputs: labor, capital, raw materials, energy
- Output: the amount of goods or services produced
- Production technology: combination of inputs to produce goods and services
Production process: bus company:
Inputs:
Labor (nr. Workers)
Capital (nr. Buses)
Energy
Output:
Bus kilometers or Passenger kilometers
Short Run vs Long Run in Production
- Short run: Some variable (e.g., labour) and some fixed (e.g., capital) inputs
The short run describes the period in which firms may adapt their production
by changing some variable factors (labor or raw material).
-> some variable (L) and some fixed (C) inputs - Long run: All inputs are variable
The long run describes the period which is long enough for firms to adapt
their production by changing all factors including capital.
-> all inputs variable
Marginal Product Definition
The additional output produced when one input increases by 1 unit, while other inputs remain constant.
Diminishing Marginal Product
- As the quantity of an input increases, its marginal product declines (other inputs are fixed).
- Graph: Represented as a concave production curve.
Cobb Douglas Production Function
Formula: Q = z Cα Lβ
- This function is widely used to represent production processes.
- z, α, β are constants; C (capital) and L (labour) are variable inputs.
This production function is commonly used by economists, because it is
capable of “illustrating” some attractive results of production theory but also
represent some production processes.
Labor Productivity Definition
- Ratio of total output to the amount of labour used.
- Example: Q/L (Total output/Total labor)
Total Factor Productivity (TFP)
- Ratio of total output to an index of total input.
- Used to compare efficiency across firms (country) or over time (two different years).
TFP = tot. Output/ tot. Input index
The weights used in these indicators are usually cost shares in the input index (factor).
Total-Factor Productivity (TFP) important issues:
TFP can differ between firms at one point in time for the following 2 reasons:
- Cost efficiency
- Scale efficiency differences
Moreover, TFP can differ between firms over time (ceteris paribus) for an
additional reason:
- Technical change (frontier shift)
Törnqvist TFP Change Index
The Törnqvist index measures the overall growth rate using revenue and cost shares as weights for inputs and outputs.
(Total-Factor Productivity)
Supply Function
Behind the supply function:
Production function
Cost function (marginal cost
function)
total product vs. marginal product of a production factor (MP)
The total product measures the total amount of output produced in physical
units of measurement (tons, kWh, …)
vs.
The marginal product of a production factor (MP) is defined by the additional
output produced as one input is increased by 1 unit (other production factors
held constant). Marginal product for labor (MP) = ΔOutput/ΔLabour Input
Using calculus (delta=partial derivative)
Total product fct. vs. marginal product fct.
Marginal product fct. is the derivative of the Total product fct.
Long Run Case: Production with two variable Inputs, Labor (L), and Capital (C)
Zuerst (links): Increasing return to scale (Increasing
Production function), dann (rechts): Decreasing
return to scale (Decreasing Production function)
Partial Productivity Indicators
A partial productivity measure relates a firm’s output to a single input factor.
A partial productivity indicator can be defined as the ratio of final output produced to a single input factor used in the production process.
For instance, labor productivity index can be defined as:
Labor Productivity (LP) = Q/L
LP1: Tot kilometres /Tot Number of Employees
LP2: Tot Passengers /Tot Number of Employees
Total Factor Productivity Change
Approach
The approaches to the TFP growth can be divided into two groups:
1. Price-based index numbers (=Törnqvist index and Fisher
Ideal index)
2. Indicators based on cost frontier or production frontier analysis (SFA, DEA)
Price-based index numbers are simple TFP indicators: Törnqvist index and Fisher
Ideal index are two of the most commonly used measures.
An important shortcoming of these measures is that they can only give an overall
estimate of total growth that could be driven by changes in efficiency, scale
economies as well as technical progress.
The decomposition of growth into different components is only possible using a
cost or production frontier model.
The results are, however, sensitive to the adopted weights and the measurement
units of input factors, thus difficult to interpret.
