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Homework answers / question archive / 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

Accounting

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?

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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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