A firm has two plants that produce outputs of three different goods. Its total labour force is...

Question

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, XXXXXXXXXX - ?)(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?

Robert · Accepted Answer

Let the fraction that went to first plant be x and the other plant be  
(1 – x) 
Output vector = (6 * x + 2 , -2 * x + 6, -6 * x + 10) 
For a = (5,5,7) to be present we must have 
6 * x + 2 = 5  
 x = ½ 
Putting this value of x we get  
-2 * x + 6 = 5 
-6 *x + 10 = 7 
Also  for b = (7, 5, 5) 
It must satisfy  
6 * x + 2 = 7 
 x = 5/6 
Putting this value of x we get  
-2 * x + 6 = 13/3 
-6 * x + 10 = 6 
 which is not equal to the required value of 5 so the vector b is not 
possible 
This occurs if output cannot be thrown away.

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...

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