The theory of the firm and industry equilibrium

Martin J. Osborne
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1.3 Exercises on isoquants

  1. Draw some isoquants for the production function
    F(z1, z2) = z1/2
    1     + z1/2
    2    .

    Solution

    The y-isoquant is given by
    y = z1/2
    1     + z1/2
    2    ,
    or
    z2 = (y − z1/2
    1    )2.
    It is shown in the following figure.

  2. Draw some isoquants for the production function
    F(z1, z2) = z2
    1     + z2
    2    .

    Solution

    The y-isoquant is given by
    y = z2
    1     + z2
    2    .

    It is shown in the following figure.

  3. Find the MRTS for the production function
    F(z1, z2) = z1/2
    1     + z1/2
    2    .

    Solution

    The y-isoquant is given by
    y = z1/2
    1     + z1/2
    2    ,
    or
    z2 = (y − z1/2
    1    )2
    so that
    MRTS(z1, z2) = (y − z1/2
    1    )z−1/2
    1     = (z2/z1)1/2.
    (Remember that the MRTS is the absolute value of the slope of an isoquant.)
  4. Which of the following production functions has a diminishing marginal rate of technical substitution?
    • F(z1, z2) = z1 + z2.
    • F(z1, z2) = z1/2
      1      z1/2
      2      .
    • F(z1, z2) = z2
      1       + z2
      2      .

    Solution

    • For the production function F(z1, z2) = z1 + z2, the isoquants are straight lines. Thus the MRTS is constant, not diminishing.
    • For the production function F(z1, z2) = z1/2
      1      z1/2
      2      , the isoquants are rectangular hyperbolae, hence convex to the origin. Thus the MRTS is diminishing.
    • For the production function F(z1, z2) = z2
      1       + z2
      2      , the isoquants are quarter circles. Thus the MRTS is increasing, not diminishing.