The theory of the firm and industry equilibrium

Martin J. Osborne
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2.1 Exercises on the cost function for a firm with one variable input

  1. A firm's production function is
    F(z1, z2) = min{2z1,4z2}.
    The amount z2 of input 2 is fixed at 20, and the prices of the inputs are w1 = 2 and w2 = 10. Find the firm's (short run) cost functions: VC, FC, STC, SMC, AVC, AFC, and SAC.

    Solution

    VC(y) = 2(y/2) = y if y ≤ 80.

    FC(y) = (10)(20) = 200.

    SMC(y) = 1 if y ≤ 80.

    AVC(y) = 1 if y ≤ 80.

    STC(y) = VC(y) + FC(y) = y + 200 if y ≤ 80.

    AFC(y) = 200/y.

    SAC(y) = 1 + 200/y if y ≤ 80.

    Given the production function it is impossible to produce more than 80 units, so all the costs except FC are infinite in this case.

  2. A firm's production function is
    F(z1, z2) = z11/4z21/2.
    The amount z2 of input 2 is fixed at 4, and the prices of the inputs are w1 = 2 and w2 = 4. Find the firm's (short run) cost functions: VC, FC, STC, SMC, AVC, AFC, and SAC.

    Solution

    We have TP(z1) = 2z11/4. How much does it cost to produce y units of output? We need
    y = 2z11/4,
    or
    z1 = (y/2)4.
    So VC(y) = 2(y/2)4 = y4/8. So AVC(y) = y3/8. And FC(y) = (4)(4) = 16, so STC(y) = y4/8 + 16. Also AFC(y) = 16/y, SAC(y) = y3/8 + 16/y, and SMC(y) = y3/2.