Lecture 13: Cost, Technology, and Scale in Microeconomics

Short-Run Cost Structures

• The Short-Run Total Cost Function: In the short run, capital is typically fixed while labor is variable. The total cost function is expressed as:     $TCS(q) = rK + wL(q)$

  • $K$ represents capital.

  • $L$ represents labor.

  • $r$ is the price of capital.

  • $w$ represents wages.

• Average Total Cost (ATC): This is defined as the firm’s total cost divided by its output.     $\text{ATC}(q) = \frac{TCS(q)}{q}$

  • The Vertical Axis Constraint: The $\text{ATC}(q)$ curve does not intersect the vertical axis because at $q = 0$, the expression involves division by zero, making the value undefined.

Components of Short-Run Costs

• Decomposition of ATC: The average total cost can be broken down into two distinct parts based on fixed and variable inputs:     $\text{ATC}(q) = \frac{FC + VC(q)}{q} = \frac{FC}{q} + \frac{VC(q)}{q}$

• Average Fixed Cost (AFC): This is the total fixed cost per unit of output.     $\text{AFC} = \frac{FC}{q}$

  • Since $FC$ is constant, $\text{AFC}$ declines continuously as output ($q$) increases.

  • This is also undefined at $q = 0$.

• Average Variable Cost (AVC): This is the variable cost per unit of output.     $\text{AVC}(q) = \frac{VC(q)}{q}$

• Visualizing the Relationship:

  • For low levels of $q$, the shape of the $\text{ATC}$ curve is primarily driven by the AFC effect (the spreading of fixed costs over more units).

  • For high levels of $q$, the AVC effect dominates the shape of the $\text{ATC}$ curve.

Marginal Cost (MC)

• Definition: Marginal cost is the increase in total cost resulting from the production of exactly one additional unit of output.

• Formal Expression:     $\text{MC}(q) = \frac{dTCS(q)}{dq}$

• The marginal cost curve is essential for understanding how costs respond to incremental changes in production scale.

Cost Derivation Worked Example

• Initial Function: Consider a firm with a short-run total cost function:     $TCS(q) = 32 + \frac{q^2}{16}$

• Derived ATC: To find the average total cost, divide the total cost by $q$:     $\text{ATC}(q) = \frac{32}{q} + \frac{q}{16}$

• Derived MC: To find the marginal cost, take the derivative of the total cost with respect to $q$:     $\text{MC}(q) = \frac{2q}{16} = \frac{q}{8}$

The Production Function and Diminishing Marginal Product

• The Production Function: This function identifies the highest possible output a firm can achieve given specific quantities of inputs. In the short run (where capital is fixed), output is a function of labor:     $q = f(L)$

• Efficiency and Feasibility:

  • Points exactly on the production function curve denote technical efficiency.

  • Points below the curve (e.g., Point A in visual models) are inefficient.

  • Points above the curve (e.g., Point B in visual models) are not feasible given current technology and inputs.

• Example Short-Run Production Function:     $q = 4L^{0.5}$

• Marginal Product (MP): This is the increase in output obtained from one additional unit of input.

• Diminishing Marginal Product: This occurs when an additional unit of input adds less to total output than the previous unit. In the context of fixed capital, each additional worker eventually contributes less to total output than those hired before them.

• Connection to Costs:

  • Output is always increasing in $L$, but the rate of increase slows down.

  • The firm experiences diminishing marginal product in the upward-sloping portion of the $\text{ATC}$ curve.

Costs in the Long Run

• Defining the Long Run: The long run is not a fixed calendar duration. It is a time horizon where all factors of production are variable. Firms can adjust both labor and capital.

• Long-Run Total Cost (TC):     $\text{TC}(q) = rK(q) + wL(q)$

  • In the long run, capital ($K$) becomes a function of output ($q$), meaning $rK(q)$ is now considered a variable cost.

  • While fixed factors of production don't exist in the long run, specific fixed costs can still exist.

• Comparison with Short Run: Since all factors are variable, there is no $\text{AFC}$ in the long run. Therefore:     $AC = AVC$

• The AC and MC Relationship:

  • The Marginal Cost ($MC$) curve always intersects the Average Cost ($AC$) curve at the minimum point of the $AC$ curve.

  • This mathematical property is a critical "solution tip" for calculating the most cost-efficient scale of production.

Returns to Scale (RTS)

• Definition: Returns to scale describes the change in output resulting from an increase in all inputs by the same proportion.

• Temporal Context: Returns to scale is only discussed in the long run because it requires scaling all inputs simultaneously. Diminishing marginal product is a short-run concept because it involves changing only one input while others are held fixed.

• Increasing Returns to Scale (IRS): Output increases more than proportionately to the increase in inputs.

  • Example ($r = w = 1$):

    • $L=1, K=1, q=1 \rightarrow TC=2, AC=2$

    • $L=2, K=2, q=2.8 \rightarrow TC=4, AC=1.4$

    • $L=4, K=4, q=8 \rightarrow TC=8, AC=1$

    • $L=8, K=8, q=22.6 \rightarrow TC=16, AC=0.7$

  • Cost Implication: Increasing returns to scale imply declining average costs.

• Decreasing Returns to Scale (DRS): Output increases less than proportionately to the increase in inputs.

  • Example ($r = w = 1$):

    • $L=1, K=1, q=1 \rightarrow TC=2, AC=2$

    • $L=2, K=2, q=1.7 \rightarrow TC=4, AC=2.4$

    • $L=4, K=4, q=2.8 \rightarrow TC=8, AC=2.8$

    • $L=8, K=8, q=4.8 \rightarrow TC=16, AC=3.3$ (Rounded from 3.33)

  • Cost Implication: Decreasing returns to scale imply increasing average costs.

• Constant Returns to Scale (CRS): Output increases in exact proportion to the increase in inputs, resulting in a flat $\text{AC}(q)$ curve.

Real-World Application: Apple’s Global Value Chain

• Sourcing Strategy: Apple sources inputs from many different countries to optimize its global value chain.

• Utilization of Scale: If there are increasing returns to scale in the production of specific iPhone components, Apple can encourage suppliers to specialize.

• Competitive Advantage: Increasing scale leads to lower average costs ($AC$). If these savings are passed through to prices, it significantly reduces Apple's sourcing costs.

Additional Materials and Discussion

• Taxation and Labor: Studies on how taxes affect casual workers suggest the impact is often attenuated (weakened) in the long run. This occurs because universities and firms can vary the number of permanent workers to compensate for cost changes.

• Global Supply Chains: Tariff wars are currently reshaping global supply chains.

• Ethical/Practical Implications of Offshoring: While moving production to cheaper destinations may lower consumer prices, these prices often fail to reflect the "true" cost (social, environmental, or systemic costs) of production.