The theory of the firm and industry equilibrium

Martin J. Osborne
Previous section Previous page Contents Text Exercises Search Next page Next section

2.2 Exercises on allocating output between two plants

  1. A firm produces output in two plants. The firm's variable cost functions in the two plants are
    VC1(y1) = 50y1
    VC2(y2) = (5/2)y2
    2    .
    The firm wants to produce 100 units of output. How should it allocate this output between the two plants in order to minimize its cost of production?

    Solution

    The marginal cost functions in the two plants are
    SMC1(y1) = 50
    SMC2(y2) = 5y2.
    Thus if the cost-minimizing allocation calls for the firm to produce a positive output in each plant, the firm wants to choose y1 and y2 such that y1 + y2 = 100 and
    50 = 5y2.
    These equations have a solution
    y1 = 90, y2 = 10.
    Thus the cost-minimizing allocation of output is y*1 = 90, y*2 = 10.
  2. A firm produces output in two plants. The firm's variable cost functions in the two plants are
    VC1(y1) = 50y1
    VC2(y2) = 100y2 + y2
    2    .
    The firm wants to produce 100 units of output. How should it allocate this output between the two plants in order to minimize its cost of production?

    Solution

    The marginal cost functions in the two plants are
    SMC1(y1) = 50
    SMC2(y2) = 100 + 2y2.
    Thus if at the cost-minimizing allocation the firm produces a positive output in each plant, it chooses y1 and y2 such that y1 + y2 = 100 and
    50 = 100 + 2y2.
    No value of y2 satisfies this condition: even if y2 = 0, the marginal cost in plant 2 is higher than than in plant 1. Thus the cost-minimizing allocation is y*1 = 100 and y*2 = 0.