Unit 3: Production and Cost
Production Functions
A production function is a mathematical equation that shows the technical relationship between physical inputs (like labor and capital) and the maximum physical output that can be produced from them, given the current state of technology.
Q = f(L, K)
Where:
- Q = Maximum quantity of output
- L = Units of Labor
- K = Units of Capital
- f = denotes the functional relationship (technology)
Short Run and Long Run
In economics, the "run" is not a specific length of time, but a conceptual period based on the flexibility of inputs.
Short Run
- Definition: The short run is a period of time in which at least one factor of production is fixed.
- Typically, Capital (K) (like the factory, machines) is considered fixed.
- Production can only be increased by adding more variable factors, like Labor (L).
- The law that applies here is the Law of Variable Proportions.
Long Run
- Definition: The long run is a period of time in which all factors of production are variable.
- The firm can change its factory size, build new plants, and adopt new technology. There are no fixed inputs.
- The law that applies here is the Returns to Scale.
Law of Variable Proportions (Short Run)
This law is also known as the "Law of Diminishing Marginal Returns."
The Law of Variable Proportions states that in the short run, as we increase the quantity of a variable input (Labor) while keeping other fixed inputs (Capital) constant, the Marginal Product (MP) of the variable input will eventually decline.
Key Concepts:
- Total Product (TP): Total quantity of output produced.
- Average Product (AP): Total Product per unit of variable input. AP = TP / L
- Marginal Product (MP): The additional output produced by using one more unit of the variable input. MP = ΔTP / ΔL
The Three Stages of Production
This law is illustrated by three stages of production, as shown in the diagram below.
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| Stage |
Description |
Total Product (TP) |
Marginal Product (MP) |
Why? |
| Stage 1: Increasing Returns |
From origin to where AP is maximum. |
Increases at an increasing rate, then at a decreasing rate. |
Increases, reaches max, then starts to fall. (MP > AP). |
Better use of fixed factor, specialization of labor. |
| Stage 2: Diminishing Returns |
From where AP is max to where MP is zero. |
Increases at a diminishing rate, reaches its maximum. |
Continues to fall, becomes zero. (MP < AP). |
Overcrowding of variable factors on the fixed factor. |
| Stage 3: Negative Returns |
After MP becomes zero. |
Starts to fall. |
Becomes negative. |
Too much variable factor; they get in each other's way. |
Exam Tip: A rational producer will always operate in Stage 2 (Diminishing Returns). They will not stop in Stage 1 (as they can still increase output efficiently) and will never operate in Stage 3 (as total output is falling).
Returns to Scale (Long Run)
This law explains the behavior of output when all inputs are changed by the same proportion (i.e., the "scale" of the firm changes) in the long run.
- 1. Increasing Returns to Scale (IRS):
- Definition: When a proportional increase in all inputs leads to a more than proportional increase in output.
- Example: 10% increase in L and K → 15% increase in Q.
- Reason: Specialization, division of labor, use of large-scale machinery (economies of scale).
- 2. Constant Returns to Scale (CRS):
- Definition: When a proportional increase in all inputs leads to an exactly proportional increase in output.
- Example: 10% increase in L and K → 10% increase in Q.
- Reason: Economies of scale have been exhausted, but diseconomies have not yet started.
- 3. Decreasing Returns to Scale (DRS):
- Definition: When a proportional increase in all inputs leads to a less than proportional increase in output.
- Example: 10% increase in L and K → 5% increase in Q.
- Reason: Managerial difficulties, coordination problems, bureaucracy (diseconomies of scale).
Iso-quant and Iso-cost Lines
Iso-quant (IQ)
- Definition: An Iso-quant (from "iso" meaning equal and "quant" meaning quantity) is a curve that shows all the possible combinations of two inputs (e.g., Labor and Capital) that produce the same level of output.
- It is also called an "equal-product curve" or "production indifference curve."
- An Iso-quant map shows a set of iso-quants, where higher curves represent higher levels of output (e.g., IQ2 > IQ1).
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Properties of Iso-quants:
- They are downward sloping from left to right.
- They are convex to the origin (due to diminishing Marginal Rate of Technical Substitution - MRTS).
- Two iso-quants never intersect each other.
- A higher iso-quant represents a higher level of output.
MRTS: The slope of the iso-quant is the Marginal Rate of Technical Substitution (MRTS). It shows the rate at which one input (K) can be substituted for another (L) while keeping output constant.
MRTS_LK = (Change in K) / (Change in L) = MP_L / MP_K
Iso-cost Line
- Definition: An Iso-cost line shows all the combinations of two inputs (Labor and Capital) that a firm can purchase with a given budget (Total Cost) and given input prices (Wage and Rent).
- It is the producer's "budget line."
- Equation: (P_L × L) + (P_K × K) = TC
Where P_L = Wage (w) and P_K = Price of Capital (r).
So, (w × L) + (r × K) = TC
- The slope of the iso-cost line is the ratio of the input prices: - (w / r).
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Producers Equilibrium
Producer's Equilibrium (or the "least-cost combination") is the point where a firm produces the maximum possible output for a given cost OR produces a given level of output at the minimum possible cost.
This equilibrium occurs at the point of tangency between an iso-quant and an iso-cost line.
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At the point of tangency (E), two conditions are met:
- The slope of the Iso-quant = The slope of the Iso-cost line.
MRTS_LK = w / r
- This can also be rewritten as:
MP_L / MP_K = w / r
or
MP_L / w = MP_K / r
This means the marginal product per dollar spent on labor is equal to the marginal product per dollar spent on capital.
Cost of Production - Types
Cost refers to the expenditure incurred by a firm on the factors of production.
- Explicit Costs (Accounting Costs): Direct, out-of-pocket payments made to outsiders for resources.
Examples: Wages paid to employees, rent for a building, cost of raw materials.
- Implicit Costs (Opportunity Costs): The value of self-owned or self-employed resources that are not paid for directly. It's the income the owner could have earned in the next-best alternative.
Examples: The salary an entrepreneur could have earned working elsewhere, the rent from their own building if they leased it out.
- Economic Cost vs. Accounting Cost:
- Accounting Cost = Explicit Costs only.
- Economic Cost = Explicit Costs + Implicit Costs.
- Fixed Costs (FC) (Short Run): Costs that do not change with the level of output. They must be paid even if output is zero.
Examples: Rent, insurance, interest on loans, salaries of permanent staff.
- Variable Costs (VC) (Short Run): Costs that do change directly with the level of output. They are zero when output is zero.
Examples: Raw materials, wages of temporary workers, electricity.
Short Run Cost Curves
In the short run, we analyze costs based on Fixed and Variable inputs.
Total Costs:
- Total Fixed Cost (TFC): A horizontal line, as it's constant regardless of output (Q).
- Total Variable Cost (TVC): Starts from the origin and rises. It is inverse-S shaped due to the Law of Variable Proportions (first increases at a decreasing rate, then an increasing rate).
- Total Cost (TC): TC = TFC + TVC. It has the same shape as TVC, but starts from the TFC level (not the origin).
Average and Marginal Costs (Per-Unit Costs):
- Average Fixed Cost (AFC): AFC = TFC / Q. It continuously falls as output increases (a shape called a "rectangular hyperbola").
- Average Variable Cost (AVC): AVC = TVC / Q. It is U-shaped due to the Law of Variable Proportions.
- Average Total Cost (ATC): ATC = TC / Q or ATC = AFC + AVC. It is also U-shaped.
- Marginal Cost (MC): The additional cost of producing one more unit of output.
MC = ΔTC / ΔQ (or ΔTVC / ΔQ).
It is also U-shaped and is the "driver" of the other average curves.
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Key Relationships (Exam Focus):
- The MC curve cuts both the AVC and ATC curves at their minimum points.
- When MC < AC (or AVC), AC (or AVC) is falling.
- When MC > AC (or AVC), AC (or AVC) is rising.
- When MC = AC (or AVC), AC (or AVC) is at its minimum.
- The vertical distance between ATC and AVC is the AFC (which gets smaller as Q increases).
Long Run Cost Curves
In the long run, all inputs are variable (no fixed costs). The firm can choose the "scale" or factory size (represented by different Short Run Average Cost curves - SRACs).
Long Run Average Cost (LRAC) Curve
- Definition: The LRAC curve shows the minimum possible average cost for producing any given level of output when the firm is free to adjust its scale.
- Derivation: It is derived as an "envelope curve" that is tangent to all the possible SRAC curves. It is *not* the sum of the minimum points of the SRACs.
- The LRAC is also U-shaped, but for different reasons than the SRAC.
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Why is the LRAC U-shaped?
- Falling Portion (Economies of Scale): The LRAC falls initially due to Economies of Scale (which are related to Increasing Returns to Scale). These are the cost advantages of becoming larger.
- Types: Technical (better machines), Managerial (specialist managers), Financial (cheaper loans), Marketing (bulk buying).
- Rising Portion (Diseconomies of Scale): The LRAC eventually rises due to Diseconomies of Scale (related to Decreasing Returns to Scale). These are the cost disadvantages of becoming *too* large.
- Types: Managerial (coordination problems), bureaucracy (red tape), communication breakdowns, low worker morale.