Production Flashcards
The Production Function
Definition:
Key Components:
Mathematical Representation:
Definition:
The production function is a mathematical representation that defines the maximum amount of output that can be produced with a given set of inputs.
Key Components:
Output (Q):
- The total level of production.
Capital Input (K):
- The quantity of capital used in production.
Labor Input (L):
- The quantity of labor employed.
Mathematical Representation:
The production function can be expressed as: Q = f(K, L)
Short-Run versus Long-Run Decisions: Fixed and Variable Inputs
Definitions?
Short-run
Period of time where some factors of production (inputs) are fixed, and constrain your choices in making input decisions.
Long-run
Period of time over which all factors of production (inputs) are variable, and can be adjusted by a manager.
Short-Run versus Long-Run Decisions: Fixed and Variable Inputs
How do you illustrate for short run decisions?
How may the production be written as?
To illustrate, suppose the level of capital is fixed in the short run. The manager can only decide how much labor to utilize. The production function may be written as :
The Production Function
Even though we call the inputs�
However, what do most production processses involve?
- Although we call the inputs capital and labor, the general ideas presented here are valid for any two inputs.
- However, most production processes involve machines of some sort (referred to by economists as capital) and people (labor), and this terminology will serve to solidify the basic ideas.
Long-run
How long is the long-run?
It depends.
For example, if it takes a company three years to acquire additional capital machines, the long run for its management is three years, and the short run is less than three years
Measures of Productivity
Total product (TP)
- Maximum level of output that can be produced with a given amount of inputs.
Average product (AP)
- A measure of the output produced per unit of input.
- Average product of labor: γπ΄πγ_πΏ=π/πΏ - Average product of capital: γπ΄πγ_πΎ=π/πΎ
Marginal product (MP)
- The change in total product (output) attributable to the last unit of an input.
- Marginal product of labor: γππγ_πΏ=βπ/βπΏ - Marginal product of capital: γππγ_πΎ=βπ/βπΎ
Relation between Productivity Measures in Action
Check ReCap
The Managerβs Role in the Production Process Example
Assume the output produced by a firm can be sold in a market at a price of Β£3;
Assume each unit of labor costs Β£400;
How may units of labor should the manager hire to maximize profits?
- We must first determine the benefit of hiring an additional worker.
- Each worker increases the firmβs output by his/her marginal product, and this increase in output can be sold in the market at a price of Β£3.
- Thus, the benefit to the firm from each unit of labor is γΒ£3ΓππγπΏ. This number is called the value marginal product of labor.
γπππγπΏ=πΓγππγπΏ (see picture)
Managerβs Role in the Production Process in Action
Profit-Maximizing Input Usage:
The manager should continue to employ labor up to the point when γπππγπΏ=π€ in the range of diminishing marginal product;
To maximize profits in the short run, the manager will hire Labor until the value of the marginal product of labor equals the wage rate: γπππγπΏ=π€
so approx. L = 9 in Table 5.2
Commonly used production function forms: (3)
Explain what they are and what type of isoquant they have.
Linear:
- A production function that assumes a perfect linear relationship between all inputs and total output
- linear isoquant
Leontief:
- A production function that assumes that inputs are used in fixed proportions
- L-shaped isoquant
Cobb-Douglas:
- A production function that assumes some degree of substitutability among inputs
- smooth convex isoquant
Commonly used algebraic production function forms: (3)
see picture
Algebraic Forms of Production: Functions in Action (picture)
see picture
Algebraic Measures of Productivity
Given the commonly used algebraic production function forms, we can compute the measures of productivity as follows:
Linear? (2)
Cobb-Douglas? (2)
see picture
Isoquants and Marginal Rate of Technical Substitution
What are isoquants?
What is the Marginal rate of technical substitutions (MRTS)
Isoquants graphically represent the various combinations of inputs K and L which yield a given level of output (Q).
Marginal rate of technical substitutions (MRTS)
- The rate at which a producer can substitute between two inputs and maintain the same level of output.
- Absolute value of the slope of the isoquant. (see picture)
Isoquants β special cases (pictures)
Linear?
Leontief?
see picture
Diminishing Marginal Rate of Technical Substitution in Action (picture)
see picture
Law of diminishing marginal rate of technical substitutions (MRTS)
A property of a production function stating that as less of one input is used, increasing amounts of another input must be employed to produce the same level of output.
Isocost and Changes in Isocost Lines
What is an isocost?
Changes in isocosts? (2)
Isocost
- Combination of inputs that yield cost the same cost.
π€πΏ+ππΎ=πΆ (see picture) - or, re-arranging to the intercept-slope formulation:
πΎ=πΆ/πβπ€/π πΏ (see picture)
Changes in isocosts
- For given input prices, isocosts farther from the origin are associated with higher costs.
- Changes in input prices change the slopes of isocost lines.
Isocost Line (picture)
see picture
Changes in the Isocost Line (picture)
Input bundles
see picture
Changes in the Isocost Line (picture)
Wage rate
see picture
Cost-Minimization Input Rule in Action (picture)
see picture
Cost minimization definition
Cost-minimizing input rule (2 with formulas)
Cost minimization
- Producing Q at the lowest possible cost.
Cost-minimizing input rule
- A firm should employ inputs such that the marginal rate of technical substitution equals the ratio of input prices (see p1)
- Produce at a given level of output where the marginal product per dollar spent is equal for all inputs: (see p2)
Long Run Expansion Path (picture)
see picture
Short Run Expansion Path (picture)
- To increase output from Q1 to Q2:
- In LR choose K2,L2 β the cost minimising point
- In SR, K is fixed so can only increase L β so choose K1, L3
- This corresponds to a higher isocost line (orange isocost vs green isocost) so producing Q2 is less efficient (more costly) in the SR.
