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Homework answers / question archive / A firm's average cost (AC) per unit of output depends on the number of hours of skilled labor (S) and the number of hours of unskilled labor (U) employed in the production process
A firm's average cost (AC) per unit of output depends on the number of hours of skilled labor (S) and the number of hours of unskilled labor (U) employed in the production process. The firm's AC function is defined below. Determine the quantities of the two inputs that will result in production at the lowest possible average cost and determine the associated minimum average cost.
The AC function is minimum when first order derivative of AC function is equal to zero and second order partial derivative of AC curve is positive for S and U respectively.
[Math Processing Error]AC=3S22+2U22−2SU−2S−6U+20dACdS=6S2−2U−2=06S2−2U=2S2=6+2U2S2=3+UdACd2S=6>0
Hence the AC is minimum at given equation of S2.
Where hours of skilled labor (S)
[Math Processing Error]dACdU=4U2−2S−6=04U2−2S=64U2=6+2SU2=1.5+0.5SdACd2U=4>0
Where hours of unskilled labor (U)
Hence the AC is minimum at given equation of U2.
Substituting S in equation for U,
[Math Processing Error]U2=1.5+0.5S2U2=1.5+0.5(3+U2)=3+0.5U2U2−0.5U2=3U2=6S2=3+6=9
Hence we get quantity of skilled labor as 9 hours and of unskilled labor is 6 hours, where Average cost is minimum.
The minimum average cost is:
[Math Processing Error]AC=3(9)2+2(6)2−2(9)(6)−2(9)−6(6)+20=173
The minimum average cost is 173 per unit.
Every unit of production has certain amount of cost attached it. It includes the inputs cost and other fixed costs of business. Such unit costs are average costs. Average costs provide minimum pints, where production is minimum in the long run. This point of output is efficient scale.
