Derivation of Short run and Long run Cost curves

In the short run, at least one factor of production is fixed (usually capital), while other factors (like labor) are variable. Costs in the short run can be divided into:

1. Fixed Costs (FC): Costs that do not change with output. Examples: rent, salaries of permanent staff.

   • Graphically: Horizontal line because it remains constant regardless of output.

2. Variable Costs (VC): Costs that change with output, such as raw materials and labor.

   • Graphically: Initially, VC rises slowly due to increasing marginal returns, then sharply due to diminishing returns.

3. Total Cost (TC): Sum of fixed and variable costs:

   TC = FC + VC

4. Average Costs:

   • Average Fixed Cost (AFC):

AFC = FC / Q​

Declines continuously as output increases (spreading effect).

• • Average Variable Cost (AVC):

    AVC = VC / Q​

Falls initially, reaches a minimum, and then rises due to diminishing returns.

• • Average Total Cost (ATC):

    ATC = TC / Q = AFC + AVC

    U-shaped, as it combines AFC and AVC.

5. Marginal Cost (MC):

• • The additional cost of producing one more unit of output:

    MC = ΔTC / ΔQ = ΔVC / ΔQ​

  • Initially falls, reaches a minimum, then rises due to diminishing marginal returns.

  • MC curve always intersects AVC and ATC at their minimum points.

Graphical Representation of Short-Run Costs:

• AFC Curve: Downward sloping.

• AVC Curve: U-shaped.

• ATC Curve: U-shaped, above AVC.

• MC Curve: Cuts AVC and ATC at their minimum points.

Derivation of Long-Run Cost Curves:

In the long run, all factors of production are variable, and firms can adjust plant size to minimize cost. Long-run cost analysis focuses on economies and diseconomies of scale.

1. Long-Run Total Cost (LRTC):

   • Represents the cost of producing any output when all inputs are variable.

   • Derived from the envelope of short-run total cost curves for different plant sizes.

   • Mathematically, LRTC is tangent to all short-run total cost curves.

2. Long-Run Average Cost (LRAC):

   LRAC = LRTC / Q​

   • U-shaped due to economies of scale (cost per unit falls as output increases) and diseconomies of scale (cost per unit rises after a certain point).

   Phases:

   • Economies of Scale: LRAC falls as output rises (specialization, division of labor, better machinery).

   • Constant Returns to Scale: LRAC is constant as output rises.

   • Diseconomies of Scale: LRAC rises due to coordination problems, inefficiencies.

3. Long-Run Marginal Cost (LRMC):

   LRMC = ΔLRTC / ΔQ​

   • Cuts LRAC at its minimum point.

Graphical Representation of Long-Run Costs:

• The LRAC curve is U-shaped and envelopes all short-run ATC curves, showing the lowest possible cost for each output level.

• LRMC curve intersects LRAC at the minimum point.