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 combination of fixed costs (TFC) and variable costs (TVC). Unlike the long run, where all factors of production are variable, in the short run, at least one factor (usually capital) remains fixed. The derivation involves understanding the relationship between total cost (TC), average cost (AC), and marginal cost (MC).

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1. Definitions of Cost Components

Before deriving the STC curve, it is essential to define the key cost components:

• Total Fixed Cost (TFC): Costs that do not change with the level of output (e.g., rent, insurance, depreciation of machinery).
• Total Variable Cost (TVC): Costs that vary with the level of output (e.g., wages, raw materials, fuel).
• Total Cost (TC): The sum of TFC and TVC. $$TC=TFC+TVC$$
• Average Fixed Cost (AFC): TFC per unit of output. $$AFC=\frac{TFC}{Q}$$
• Average Variable Cost (AVC): TVC per unit of output. $$AVC=\frac{TVC}{Q}$$
• Average Total Cost (ATC): TC per unit of output. $$ATC=\frac{TC}{Q}=AFC+AVC$$
• Marginal Cost (MC): The additional cost of producing one more unit of output. $$MC=\frac{\Delta TC}{\Delta Q}=\frac{\Delta TVC}{\Delta Q}$$

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2. Shape of the Short-Run Total Cost (STC) Curve

The STC curve is derived by summing TFC and TVC at different levels of output. Its shape depends on the behavior of TVC and TFC:

a) Total Fixed Cost (TFC) Curve

• The TFC curve is a horizontal straight line parallel to the X-axis because fixed costs do not change with output.
• Example: If a firm’s rent is Rs. 10,000 per month, TFC remains Rs. 10,000 regardless of output.

b) Total Variable Cost (TVC) Curve

• The TVC curve starts from the origin (0,0) and rises at an increasing rate due to the law of diminishing marginal returns.
• Initially, TVC increases at a decreasing rate (due to specialization and efficiency gains).
• Later, TVC increases at an increasing rate (due to overutilization of fixed factors, leading to higher per-unit costs).

c) Total Cost (TC) Curve

• The TC curve is obtained by vertically adding TFC and TVC at each level of output.
• Since TFC is constant, the TC curve is parallel to the TVC curve but shifted upward by the amount of TFC.
• The shape of TC is initially convex (concave upward) and then convex (concave downward), reflecting the behavior of TVC.

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3. Derivation of Average Cost Curves from TC

The average cost curves (AFC, AVC, ATC) are derived from the TC curve and help explain the shape of the STC curve.

a) Average Fixed Cost (AFC) Curve

• AFC is calculated as $\frac{TFC}{Q}$.
• Since TFC is constant, AFC declines continuously as output increases (due to the spreading effect).
• The AFC curve is downward-sloping and asymptotically approaches the X-axis (never touches it).

b) Average Variable Cost (AVC) Curve

• AVC is calculated as $\frac{TVC}{Q}$.
• Initially, AVC declines due to increasing returns (specialization of labor).
• After a certain point, AVC rises due to diminishing returns (overuse of fixed factors).
• The AVC curve is U-shaped.

c) Average Total Cost (ATC) Curve

• ATC is the sum of AFC and AVC: $ATC=AFC+AVC$.
• Since AFC is always declining and AVC is U-shaped, ATC is also U-shaped.
• The minimum point of ATC determines the least-cost output for the firm.

d) Marginal Cost (MC) Curve

• MC is the slope of the TC curve (or TVC curve, since $\Delta TC=\Delta TVC$).
• The MC curve cuts the ATC curve at its minimum point.
• The MC curve is U-shaped due to the law of diminishing returns.

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4. Relationship Between MC and ATC

The MC curve intersects the ATC curve at its lowest point. This is because:

• When MC < ATC, ATC is falling (additional units reduce the average cost).
• When MC > ATC, ATC is rising (additional units increase the average cost).
• The minimum point of ATC occurs where MC = ATC.

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5. Numerical Example (Derivation of STC, ATC, MC)

Assume a firm has:

• TFC = Rs. 10,000 (fixed)
• TVC at different output levels (given in the table below):

Key Observations:

1. TC increases at a decreasing rate first, then at an increasing rate (due to diminishing returns).
2. AFC continuously declines as output increases.
3. AVC first declines, reaches a minimum, and then rises (U-shaped).
4. ATC is U-shaped and its minimum occurs where MC = ATC (at Q = 6).
5. MC cuts ATC at its lowest point, confirming the theoretical relationship.

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6. Why is the STC Curve U-Shaped?

The U-shape of the STC curve is explained by:

1. Initially decreasing slope (concave upward):

   • Due to increasing returns to a factor (e.g., better utilization of fixed capital).
   • MC is falling, so each additional unit adds less to total cost than the previous one.

2. After a certain point, increasing slope (convex upward):

   • Due to diminishing returns (overuse of fixed factors).
   • MC is rising, so each additional unit adds more to total cost than the previous one.

3. The minimum point of STC:

   • The lowest point of the STC curve corresponds to the minimum efficient scale (MES) of production.
   • Beyond this point, economies of scale turn into diseconomies of scale.

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7. Practical Implications

• Firms aim to produce at the minimum point of ATC for cost efficiency.
• If MC > ATC, the firm should reduce output to minimize costs.
• If MC < ATC, the firm should increase output to lower average costs.
• The short-run cost curves help in profit maximization and shutdown decisions.

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The derivation of the short-run total cost curve is fundamental in microeconomics, explaining how firms make production and pricing decisions under constraints. The U-shaped ATC and MC curves are direct consequences of the law of diminishing returns, which governs short-run production behavior.