Explain the derivation of short run total cost curves. [2+4+4]

Derivation of Short-Run Total Cost Curves

The short-run total cost (STC) curve is derived from the law of variable proportions and the behavior of fixed and variable costs in the short run. In the short run, at least one factor of production (usually capital) is fixed, while others (like labor) can be varied. The derivation involves understanding how changes in variable inputs affect total costs, leading to the characteristic U-shaped average cost curves.

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1. Basic Cost Concepts in the Short Run (2 marks)

In the short run, a firm’s total cost (TC) is the sum of fixed costs (TFC) and variable costs (TVC): $$TC=TFC+TVC$$

• Fixed Costs (TFC): Costs that do not change with output (e.g., rent, machinery depreciation).
• Variable Costs (TVC): Costs that vary with output (e.g., wages, raw materials).

The total cost curve (STC) is derived by adding TFC and TVC at each level of output. Since TFC remains constant, the shape of the STC curve depends entirely on the behavior of TVC.

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2. Law of Variable Proportions and Cost Behavior (4 marks)

The law of variable proportions states that as more units of a variable input (e.g., labor) are added to a fixed input (e.g., capital), the marginal product (MP) of the variable input eventually declines. This leads to three phases of production:

Phase 1: Increasing Returns (Diminishing Marginal Costs)

• Initially, adding more labor increases total output (TP) at an increasing rate.
• Marginal product (MP) is rising, so marginal cost (MC) falls.
• Average variable cost (AVC) and average total cost (ATC) decline.

Phase 2: Diminishing Returns (Rising Marginal Costs)

• As more labor is added, MP starts falling (due to overcrowding of fixed factors).
• MC begins to rise because each additional unit of labor contributes less to output.
• AVC and ATC start rising as MC exceeds AVC.

Phase 3: Negative Returns (Rising Costs)

• If labor is added beyond a certain point, TP may decline, leading to negative MP.
• MC becomes very high, causing AVC and ATC to rise sharply.

The short-run total cost (STC) curve reflects these changes:

• Initially, STC rises at a decreasing rate (due to increasing returns).
• Later, STC rises at an increasing rate (due to diminishing and negative returns).

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3. Derivation of STC from TFC and TVC (4 marks)

The STC curve is obtained by vertically adding TFC and TVC at each level of output. This can be visualized as follows:

Step 1: Plotting TFC and TVC

• TFC is a horizontal line (constant at all output levels).
• TVC starts at the origin (zero output) and rises, initially at a decreasing rate (due to increasing returns), then at an increasing rate (due to diminishing returns).

Step 2: Adding TFC and TVC to Get STC

• At zero output, $STC=TFC$ (since $TVC=0$).
• As output increases, STC rises because $TVC$ increases.
• The shape of STC depends on TVC:
  • If TVC rises slowly (increasing returns), STC rises slowly.
  • If TVC rises rapidly (diminishing returns), STC rises steeply.

Graphical Representation

• TFC remains constant at Rs. 50.
• TVC starts at 0, rises slowly at first, then steeply.
• STC is the vertical sum of TFC and TVC, showing a U-shaped pattern when combined with ATC.

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4. Relationship Between STC, ATC, AVC, and MC (Key Insight)

• STC = TFC + TVC
• ATC = STC / Q
• AVC = TVC / Q
• MC = ΔTC / ΔQ

The STC curve is steeper than the TVC curve because it includes the constant TFC. The slope of STC at any point gives the MC.

Key Observations:

1. STC always rises with output (since TVC increases).
2. STC is convex (U-shaped) because MC rises after a certain point.
3. The minimum point of STC corresponds to the minimum efficient scale of production.

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Conclusion

The short-run total cost curve (STC) is derived by adding fixed costs (TFC) and variable costs (TVC) at each level of output. Its shape is determined by the law of variable proportions, where marginal costs (MC) initially fall and then rise, leading to a U-shaped ATC curve. The STC curve is crucial for understanding a firm’s cost structure in the short run and helps in decision-making regarding production levels.