The theory of the firm and industry equilibrium

Martin J. Osborne
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1.4 Exercises on returns to scale

  1. Determine the returns to scale of the following production functions:
    1. F(z1, z2) = [z2
      1      1 + z2
      2      ]1/2.
    2. F(z1, z2) = (z1 + z2)1/2.
    3. F(z1, z2) = z1/2
      1       + z2.

    Solution

    1. F(αz1, α2) = [α2z2
      1       + α2z2
      2      ]1/2 = (α2)1/2(z2
      1       + z2
      2      )1/2 = αF(z1, z2). Thus this production function has CRTS.
    2. F(αz1, αz2) = (αz1 + αz2)1/2 = α1/2(z1 + z2)1/2 = α1/2F(z1, z2). Thus this production function has DRTS.
    3. F(αz1, αz2) = α1/2z1/2
      1       + αz2 < α(z1/2
      1       + z2) = αF(z1, z2) for α > 1. Thus this production function has DRTS.