Costs: Fixed, Variable, and Mixed Costs

Fixed Costs and Variable Costs: Overview

  • Fixed cost (FC) is a committed, time-invariant expense within the relevant range; it does not change with the level of activity.
    • Example given: FC = 10{,}000.
    • Fixed cost per unit declines as output increases because the same fixed amount is spread across more units.
    • At 1 unit: FC per unit = rac{FC}{Q} = rac{10{,}000}{1} = 10{,}000.
    • At 5 units: FC per unit = rac{10{,}000}{5} = 2{,}000.
    • At 100 units: FC per unit = rac{10{,}000}{100} = 100.
  • Total variable cost (TVC) starts at zero and increases with activity; it has a positive slope on a TVC vs. output graph. TVC can be thought of as the cost that varies with the level of output.
    • In the simplified example, at unit 1 TVC is 25, and at 100 units TVC is 500, illustrating an upward-sloping TVC curve.
    • This leads to an average variable cost (AVC) that can change with output: at Q = 100, AVC = rac{TVC}{Q} = rac{500}{100} = 5; at Q = 1, AVC = rac{25}{1} = 25.
  • Total cost (TC) is the sum of fixed and variable costs:
    • TC(Q) = FC + TVC(Q)
    • With the given numbers, TC(1) = 10{,}000 + 25 = 10{,}025 and TC(100) = 10{,}000 + 500 = 10{,}500.
  • Per-unit cost concepts (cost per unit):
    • Average fixed cost (AFC) = rac{FC}{Q}
    • Average variable cost (AVC) = rac{TVC(Q)}{Q}
    • Average total cost (ATC) = rac{TC(Q)}{Q} = rac{FC}{Q} + rac{TVC(Q)}{Q} = AFC + AVC
  • Activity level and interpretation:
    • The horizontal axis represents the activity level (output) Q; the cost curves reflect how costs behave as Q changes.
    • The fixed cost portion remains constant in total within the relevant range, but its per-unit impact diminishes as Q grows.
    • The variable cost portion rises with output (positive slope of the TVC curve).

Cost behavior and implications

  • Fixed costs are committed costs; they do not vary with the level of activity in the relevant range.
  • Variable costs depend on the level of activity; more output typically means higher total variable costs.
  • The two cost types are not perfectly separable in practice, because some costs have both fixed and variable components depending on what you’re measuring (e.g., time, space, resources).
  • The concept of the relevant range matters: within a given range of activity, FC is treated as constant in total, while FC per unit falls as activity increases; outside this range, cost behavior may change (costs can become stepwise, change slope, etc.).

Real-world cost components and examples

  • Fixed costs (examples):
    • Rent for space
    • Manager salaries (fixed component in some cases)
    • Legal retainer fees (often treated as fixed within a period)
  • Variable costs (examples):
    • Janitorial expenses that scale with usage
    • Cable TV pay-per-view or channels that incur extra charges beyond a base fee
    • Utilities or maintenance that scale with activity levels
  • Mixed (semi-variable) costs: have both fixed and variable components
    • Example: Salaries with a fixed base plus variable bonuses or incentives; total salary is a mix of fixed base and a variable component tied to performance or hours.
    • In the notes, a manager’s pay is discussed as having a fixed salary portion plus a variable bonus; the combination is a mixed cost.
    • The example notes that a banded pay structure might assign a fixed base (e.g., 84) and a variable component (e.g., bonus) that changes with output or performance.
  • A concrete salary example discussed: total server salary is described as 5 times 2,000 = 10,000, illustrating how a mix of fixed and variable elements can still lead to a total that behaves like a variable cost component when aggregated across outputs or hours.

The role of the relevant range

  • Cost behavior (fixed vs. variable) is often valid only within a given relevant range of activity.
  • Across different ranges, fixed costs may still be constant in total but could shift (e.g., step costs) or variable costs per unit may change due to capacity limits, efficiency changes, or other factors.
  • When evaluating costs for decision-making, identify the relevant range to ensure the correct interpretation of FC, TVC, and their per-unit metrics.

Graphical intuition (how the pieces fit together)

  • TVC vs. Q: TVC increases with Q; the slope is positive, reflecting more total variable costs as output rises.
  • AFC vs. Q: AFC = FC / Q; as Q increases, AFC falls, illustrating the spreading of fixed costs over more units.
  • ATC vs. Q: ATC = AFC + AVC; because AFC falls with Q while AVC can rise or fall depending on TVC behavior, ATC may fall, rise, or be U-shaped depending on the relative magnitudes of AFC and AVC.

Quick numerical recap (using the transcript’s figures)

  • Fixed cost: FC = 10{,}000
  • Output levels and fixed cost per unit:
    • Q = 1 → AFC = rac{10{,}000}{1} = 10{,}000
    • Q = 5 → AFC = rac{10{,}000}{5} = 2{,}000
    • Q = 100 → AFC = rac{10{,}000}{100} = 100
  • Total variable cost (TVC) example points:
    • TVC(1) = 25
    • TVC(100) = 500
  • Derived per-unit variable costs (AVC):
    • AVC(1) = rac{25}{1} = 25
    • AVC(100) = rac{500}{100} = 5
  • Total cost (TC) examples:
    • TC(1) = FC + TVC(1) = 10{,}000 + 25 = 10{,}025
    • TC(100) = FC + TVC(100) = 10{,}000 + 500 = 10{,}500
  • Average total cost (ATC) examples (not numerically shown in transcript, but formulaic):
    • ATC(Q) = rac{TC(Q)}{Q} = rac{FC}{Q} + rac{TVC(Q)}{Q} = AFC(Q) + AVC(Q)

Formulas to remember (LaTeX)

  • Total cost: TC(Q) = FC + TVC(Q)
  • Average fixed cost: AFC(Q) = rac{FC}{Q}
  • Average variable cost: AVC(Q) = rac{TVC(Q)}{Q}
  • Average total cost: ATC(Q) = rac{TC(Q)}{Q} = rac{FC}{Q} + rac{TVC(Q)}{Q} = AFC(Q) + AVC(Q)

Practical implications for decision making

  • When planning capacity or pricing, recognize that FC is spread over more units at higher output, reducing AFC but TC and TVC grow with output.
  • Variable costs must be anticipated for each unit increase in output; TVC is the driver of changes in TC via TVC.
  • If a cost has both fixed and variable components (mixed cost), evaluate both components separately to understand how TC responds to changes in output.
  • Always consider the relevant range, as cost behavior may shift outside the assumed range (e.g., fixed costs becoming step costs).

Summary takeaways

  • FC is fixed in total within the relevant range; FC per unit falls as output rises.
  • TVC increases with output; it starts at zero when output is zero.
  • TC is the sum of FC and TVC; ATC is the per-unit TC.
  • Mixed costs combine fixed and variable elements (e.g., base salary plus bonus).
  • Real-world examples help illustrate fixed, variable, and mixed components and their practical implications (rent, maintenance, legal fees, cable, bonuses).
  • The relevant range governs how costs behave; costs can change behavior beyond this range.