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The quantity, Q, of a certain product manufactured depends on the quantity of labor, L, and of capital, K, used according to the function Q=900L1/2K2/3Q=900L1/2K2/3
The quantity, Q, of a certain product manufactured depends on the quantity of labor, L, and of capital, K, used according to the function Q=900L1/2K2/3Q=900L1/2K2/3. Labor costs $100 per unit and capital costs $200 per unit. What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?
Expert Solution
Given Data
- The total production is Q=36000Q=36000.
The production function is given as,
Q=900L12K23Q=900L12K23
Substitute the known values in the above equation.
36000=900L12K2340K23=L12L=1600K4336000=900L12K2340K23=L12L=1600K43
The expression for the production cost.
P=L+2KP=L+2K
Here, L is the total number of labor and K is the total capital and L unit is hundreds of dollar.
Substitute the value of L in the above equation.
P=1600K43+2KP=1600K43+2K
For minimum cost, the production cost should be minimizing.
dPdK=0dPdK=0
Substitute the known value.
d1600K43+2KdK=0−1600×−43K−73+2=0800×43=K−73K=19.848≈20d1600K43+2KdK=0−1600×−43K−73+2=0800×43=K−73K=19.848≈20
Therefore the value of L is calculated as,
L=1600K43L=1600K43
Substitute the known values in the above equation.
L=16002043=29.472≈30L=16002043=29.472≈30
The production cost is calculated as,
P=L+2KP=L+2K
Substitute the known values.
P=30+2×20=70P=30+2×20=70
Thus, the production cost is $7000$7000.
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