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corie corie
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Posts: 767
4 years ago
Davy Metal Company produces brass fittings.  Davy's engineers estimate the production function represented below as relevant for their long-run capital-labor decisions.
      Q = 500L0.6K0.8,       
where Q = annual output measured in pounds,   
L = labor measured in person hours,   
K = capital measured in machine hours.       
The marginal products of labor and capital are:
      MPL = 300L-0.4K0.8    MPK = 400L0.6K-0.2       
Davy's employees are relatively highly skilled and earn $15 per hour. The firm estimates a rental charge of $50 per hour on capital.  Davy forecasts annual costs of $500,000 per year, measured in real dollars.

a.   Determine the firm's optimal capital-labor ratio, given the information above.
b.   How much capital and labor should the firm employ, given the $500,000 budget?  Calculate the firm's output.
c.   Davy is currently negotiating with a newly organized union.  The firm's personnel manager indicates that the wage may rise to $22.50 under the proposed union contract.  Analyze the effect of the higher union wage on the optimal capital-labor ratio and the firm's employment of capital and labor.  What will happen to the firm's output?
Textbook 

Microeconomics


Edition: 8th
Author:
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boransalboransal
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Posts: 474
4 years ago
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a.    
MPL = 300 L-0.4K0.8 = 300
MPK = 400 L0.6K-0.2 = 400
MRTS =   = 0.75   
MRTS = 0.75
Equate MRTS to   =  .
   0.75  = 
   0.75  = 0.3
  = 0.4; K=0.4L

b.
   C = 500,000
   C = wL + rK
   500,000 = 15L + 50K       
K = 0.4L from optimal ratio
   500,000 = 15L + 50(0.4L)
   500,000 = 15L + 20L
   500,000 = 35L    
      L = 14,285.71 or 14,286 hours       
Substitute to solve for K.
   500,000 = 15(14286) + 50K
   500,000 = 214,290 + 50K
   285,710 = 50K    
      K = 5714.20    
      or K = 5714
   Q = 500(14,286)0.6(5,714)0.8
   Q = 157,568,191    

c.
   MRTS = 0.75
   New   =   = 0.45
   Equating MRTS to   =  .
      0.75  = 0.75
        = 0.6
   K = 0.6L


   Substitute into C:
      500,000 = 22.50L + 50K    
      K = 0.60L
      500,000 = 22.50L + 50(0.6L)
      500,000 = 22.50L + 30L
      500,000 = 52.50L    
      L = 9,523.8 or 9,524       
L fell from 14,286 to 9,524.  Substitute to solve for K.
      500,000 = 22.50(9,524) + 50K
      285,710 = 50K    
      K = 5,714.20 or 5,714       
K remains constant.
      Q = 500(9524)0.6(5714)0.8    
         Q = 123,541,771.8       
Output fell from 157,568,202.5 to 123,541,771.8.
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