Cost Curves

Intermediate Microeconomics (Econ 100A)

Kristian López Vargas

UCSC - Spring 2017

Cost Curves - Average Costs

  • Total cost: $ c(q) = c_v(q) + F $

  • Average Cost: $ AC(q) = \frac{c(q)}{q} = \frac{c_v(q)}{q} + \frac{F}{q} $

  • That is: $ AC(q) = AVC(q) + AFC(q) $

  • Notice FC does not depend on $ q $, but AFC does depend on $ q $.

  • Can AFC increase in $ q $

Average Costs

Marginal Cost

  • Marginal cost is the change in cost due to change in output

  • $ c’(q) = \frac{ ∂c(q) }{ ∂q } = \frac{ ∂c_v(q) }{ ∂q } $

The geometry of total cost, AC, AVC and MC

Some relations of Cost Curves

  • Marginal cost equals AVC at zero units of output

    • Because AVC starts at the origin.
  • MC: crosses at minimum points of AC and AVC.

    • MC < AC when AC is decreasing and MC > AC when AC is increasing.

Cost Curves

Cost Curves - Example:

  • $ c(q) = 1 + q^2 $

  • $ c_v(q) = ? $

  • $ FC(q) = ? $

  • $ AVC(q) = ? $

  • $ AFC(q) = ? $

  • $ AC(q) = ? $

  • $ MC(q) = ? $

Cost Curves - Example:

Cost Curves - do-at-home examples:

  • $ c(q) = 10 - 0.5(q-2)^2 + (q-2)^3 $

  • Try with all kinds of functions...

Cost minimization in two plants

  • Possibly two technologies, therefore two cost functions $ c_1(q_1) \text{ and } c_2(q_2) $

  • Graphical approach

    • Use a production-requirement "box"
  • Mathematical approach

    • $ \text{minimize} (C = c_1(q_1) + c_2(q_2)) $ subject to: $ q_1 + q_2 = q $

    • replace $ q_2 $ by $ q-q_1 $ and solve $ dC/dq_1 = 0$

    • Solution: $ q_1 \text{ and } q_2 $ such that $ MC_1(q_1) = MC_2(q_2) $

Long-run and short-run cost function - Example.

  • $ q = 50 L^{0.5} K^{0.5} $

  • Find long run cost function:

    • Total Cost (TC): $ c(w,r,q) = (q/25)(wr)^{0.5} $

    • $ AC(w,r,q) = (1/25)(wr)^{0.5} $

    • $ MC(w,r,q) = (1/25)(wr)^{0.5} $

Long-run and short-run cost function - Example.

  • $ q = 50 L^{0.5} K^{0.5} $

  • Find short-run cost function ($ K = \bar{K} $):

    • $ L^{SR} = \frac{ q^2 }{ 50^2 \bar{K} } $

    • $ c^{SR}(w,r,q) = w \frac{ q^2 }{ 50^2 \bar{K} } + r \bar{K} $

    • $ {AC}^{SR}(w,r,q) = w \frac{ q }{ 50^2 \bar{K} } + r \frac { \bar{K} }{ q } $

    • $ {MC}^{SR}(w,r,q) = 2 w \frac{ q }{ 50^2 \bar{K} } $

If for example w = 25 and r = 100:

Short-run AC (SAC) and long-run AC (SAC)

  • Cost curves coincide if fixed level of capital is also LR solution.

Short-run AC (SAC) and long-run AC (SAC)

  • More generally: LR cost curves envelope from below the SR ones.

EconGraphs - general Cobb-Douglas