A firm has two plants that produce outputs of three different goods. Its total labour force is fixed. When a fraction ? of its labour force is allocated to its first plant and a fraction 1 - ? to its...

A firm has two plants that produce outputs of three different goods. Its total labour force is fixed. When a fraction ? of its labour force is allocated to its first plant and a fraction 1 - ? to its second plant (with 0 = ? = 1), the total output of the three different goods are given by the vector ?(8, 4, 4) + (1 - ?)(2, 6, 10) = (6? + 2, -2? + 6, -6? + 10).

(a) Is it possible for the firm to produce either of the two output vectors a = (5, 5, 7) and b = (7, 5, 5) if output cannot be thrown away?

(b) How do your answers to part (a) change if output can be thrown away?

(c) How will the revenue-maximizing choice of the fraction ? depend upon the selling prices (p₁, p₂, p₃) of the three goods? What condition must be satisfied by these prices if both plants are to remain in use?