### How will the revenue-maximizing choice of the fraction

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

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