Formulae and Relationships Between Costs

Total Cost

The total expenditure on all factors of production.

• TC = TFC + TVC

Total Fixed Cost

Costs that do not vary with the level of output (e.g., rent, insurance). Only exists in the short run.

• TFC = constant

Total Variable Cost

Costs that vary directly with the level of output (e.g., raw materials, direct labour wages).

• TVC varies with Q

Average (Total) Cost

The cost per unit of output. U-shaped.

• AC = TC / Q

Average Fixed Cost

Fixed cost per unit of output. Always declines as output ($Q$) increases (spreading the overhead).

• AFC = TFC / Q

Average Variable Cost

Variable cost per unit of output. U-shaped.

• AVC = TVC / Q

Marginal Cost

The extra cost of producing one more unit of output. U-shaped.

• MC = Change in TC / Change in Q

Derivation of Short-Run Cost Curves from the Assumption of Diminishing Marginal Productivity

The shape of the short-run cost curves (MC, AVC, and ATC) is primarily determined by the Law of Diminishing Marginal Returns (also known as Diminishing Marginal Productivity).

Law of Diminishing Marginal Returns (Short Run)

• In the short run, at least one factor of production (capital) is fixed.

• The law states that as successive units of a variable factor (labour) are added to a fixed factor, there will come a point where the marginal product (MP) of the variable factor will begin to decrease.

  • Initially, increasing marginal returns (from specialisation) cause MP to rise.

  • Eventually, diminishing marginal returns set in (e.g., too many workers sharing one machine), causing MP to fall.

Inverse Relationship between Productivity and Costs

The relationship between the output curves (Marginal Product, MP, and Average Product, AP) and the per-unit cost curves (Marginal Cost, MC, and Average Variable Cost, AVC) is inverse:

1. 1. When Marginal Product (MP) is rising (increasing returns):

   • The additional output per worker is high.

   • The cost of producing an additional unit (Marginal Cost, MC) is falling.

3. 2. When Marginal Product (MP) is falling (diminishing returns):

   • The additional output per worker is low.

   • The cost of producing an additional unit (Marginal Cost, MC) is rising.

5. 3. This inverse relationship shapes the Marginal Cost (MC) curve as U-shaped.

• The Average Variable Cost (AVC) curve follows a similar U-shape, inversely related to the Average Product (AP) curve.

• The MC curve intersects the AVC and ATC curves at their minimum points.

  • If $MC < AVC/ATC$, the average is being pulled down.

  • If $MC > AVC/ATC$, the average is being pushed up.

Relationship between Short-Run and Long-Run Average Cost Curves

The distinction between the short run and the long run is crucial here:

• Short Run (SR): At least one factor (plant/capital) is fixed. Firms adjust output by varying variable factors.

• Long Run (LR): All factors of production are variable. Firms can change their scale of operation (e.g., build a new, larger factory).

Long-Run Average Cost (LRAC) Curve

The Long-Run Average Cost (LRAC) curve (also known as the planning curve) is an envelope curve that traces the minimum average cost of production for every possible level of output, allowing all factors of production to be varied.

• Each Short-Run Average Cost (SRAC) curve represents a specific, fixed plant size or scale of operation.

• The LRAC curve is the tangent to the bottom of all possible SRAC curves.

LRAC Shape and Economies of Scale

The LRAC curve is typically U-shaped (though sometimes shown as L-shaped in reality) due to the concepts of returns to scale:

1. Increasing Returns to Scale (IRS) / Economies of Scale (EoS):

   • As the firm increases its scale of production (moves to larger SRAC curves), its LRAC is falling.

   • This is due to factors like specialisation, bulk-buying, and financial advantages.

3. Constant Returns to Scale (CRS):

   • The firm's average cost is at its minimum.

   • This is the minimum point of the LRAC curve, known as the Minimum Efficient Scale (MES).

5. Decreasing Returns to Scale (DRS) / Diseconomies of Scale (DoS):

   • As the firm continues to expand its scale, its LRAC is rising.

   • This is often due to coordination and communication problems in a vast organisation.

Key Relationship Points

• For any given output level, the LRAC will always be less than or equal to the cost on the relevant SRAC curve.

• The LRAC is an envelope because in the long run, the firm can choose the optimal plant size for any desired output level, ensuring the lowest possible average cost.