A firm’s production technology is given by the production function q = 0.25 LK where L represents labor hours, K machine hours and q the amount of output. The market wage and rental rates are, w= $16 and r = $256. The firm is operating in the long run where it can adjust both inputs.
(b) Suppose that the firm wants to produce 100 units of output. Determine the cost minimizing combination of L and K. Calculate the resulting long run total cost. Show and explain all calculations (
e) Using Excel- Solver verify your answers to (b) above. (Show your work. Show the spreadsheets in detail. Show the Solver window embedded on the relevant worksheet so that the commands in the Solver window become directly visible and are linked to the cells of the worksheet.)
q = 0.25LK
(a) Long run cost is minimized when MPL/MPK = w/r, or MPL/w = MPK/r
MPL = q/L = 0.25K
MPK = q/K = 0.25L
When cost is minimized,
MPL/MPK = K/L = 16/256 = 1/16, therefore
MPK/MPL = L/K = 16
However, current value of (L/K) = 10 (given) < 16, so firm is not minimizing cost.
To use optimal input ratio, firm has to use more labor and less capital until (MPL/MPK) = (w/r).
(b) When q = 100, we have
100 = 0.25LK
LK = 400.......(1)
Cost-minimizing condition: (MPL/MPK) = K/L = w/r = 16/256 = 1/16
L = 16K
Substituting in (1),
16K x K = 400
K2 = 400/16 = 25
K = 5
L = 16 x 5 = 80
Total cost ($) = wL + rK = 16 x 80 + 256 x 5 = 1280 + 1280 = 2560
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