Cost Curves

  • Three main cost curves
    • Total cost curve
    • Variable cost curve
    • Average total cost curve
  • Related cost curves
    • Average variable cost curve
    • Average fixed cost curve
    • Marginal cost curve
  • In all of these curves, the horizontal axis is the output level (y) and the vertical axis is the cost

Cost Functions

  • F is the cost of the short run fixed inputs (AKA fixed cost)
  • Cv(y) is the cost of the variable inputs (AKA variable cost)
  • C(y) is the total cost of all inputs (AKA total cost)
    • C(y) = F + Cv(y)

Cost Curves

  • Notice how C(y) and Cv(y) have the same slope, but C(y) is shifted up by F units

Average Cost Functions

  • Three functions: average fixed cost, average variable cost, and average total cost
  • For y > 0, the firm’s average total cost function is AC(y) = (F + Cv(y)) / y = AFC(y) + AVC(y)
    • AC and ATC are synonymous
  • As y increases, AC and AVC will get closer and closer to each other, as AFC will trend towards 0 as y increases
  • Note that natural monopolies arise as a result of decreasing AFC; the more customers a monopoly can serve, the lower the AFC gets and the lower the cost for the consumer

Marginal Cost Function

  • The marginal cost function is the rate of change of the variable cost as the output level changes
MC(y)=∂Cv(y)∂y=∂C(y)∂yThis result occurs because the total cost and variable cost functions have the same slope.MC(y) = \frac{\partial C_v (y)}{\partial y} = \frac{\partial C (y)}{\partial y} \\ \text{This result occurs because the total cost and variable cost functions have the same slope.}

Marginal and Average Cost Functions

  • Note that, if the marginal cost is lower than the average cost, then the average cost will go down, and vice versa
  • This means that the MC curve will intersect the ATC and AVC at their lowest point, being lower than them before the intersection and higher than them afterwards

Average and Marginal Curves

Mathematical proof:AVC(y)=Cv(y)y∂AVC(y)y=y⋅∂Cv(y)/∂y−Cv(y)y2 by the Quotient RuleMC=∂Cv(y)/∂y∂AVC(y)y>,<,=0 when y⋅MC(y)>,<,=Cv(y)∂AVC(y)y>,<,=0 when MC(y)>,<,=AVC(y)The same can be proven for ATC as well, where ATC(y)=C(y)y∂AVC(y)y=y⋅∂C(y)/∂y−C(y)y2MC=∂C(y)/∂y∂ATC(y)y>,<,=0 when y⋅MC(y)>,<,=C(y)∂ATC(y)y>,<,=0 when MC(y)>,<,=ATC(y)\text{Mathematical proof:} \\ AVC(y) = \frac{C_v (y)}{y} \frac{\partial AVC(y)}{y} = \frac{y \cdot \partial C_v (y) / \partial y - C_v(y)}{y^2} \text{ by the Quotient Rule} \\ MC = \partial C_v (y) / \partial y \\ \frac{\partial AVC(y)}{y} >, <, = 0 \text{ when } y \cdot MC(y) >, <, = C_v(y) \\ \frac{\partial AVC(y)}{y} >, <, = 0 \text{ when } MC(y) >, <, = AVC(y) \\ \text{The same can be proven for ATC as well, where } ATC(y) = \frac{C (y)}{y} \\ \frac{\partial AVC(y)}{y} = \frac{y \cdot \partial C (y) / \partial y - C(y)}{y^2} \\ MC = \partial C(y) / \partial y \\ \frac{\partial ATC(y)}{y} >, <, = 0 \text{ when } y \cdot MC(y) >, <, = C(y) \\ \frac{\partial ATC(y)}{y} >, <, = 0 \text{ when } MC(y) >, <, = ATC(y) \\

In other words, the MC curve will intersect the ATC curve from below at the ATC curve’s minimum. The same applies to the AVC curve.