Question

The cost of producing x units of a commodity is given by. TC(x)=100+40x+2x^2

Derive the following functions:

FC =

VC =

AFC =

AVC =

ATC =

Find TC(x+1)-TC(x), and explain in words what this means.

Answer 1

The total cost function of producing x units of good is given as:TC(x)=100+40x+2x^2

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The fixed cost (FC) is the portion of the total cost incurred independent of the goods produced, that is the cost that firm bears when x=0

That means, FC = TC(x=0) = 100

----------------------------------------------------------------------------------------------------------------------------------------------------------------------The variable cost (VC) is the portion of the total cost that is dependent on goods produced and that which changes with each unit of the good produced.

Thus, VC = 40x+2x^2

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The average fixed cost (AFC) is the fixed cost (FC) per unit output produced x

Hence, AFC = FC/x

AFC = = 100/x

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The average variable cost (AVC) is the variable cost (VC) per unit output x

Hence, AVC = VC/x = (40x+2x^2)/x

AVC = 40 + 2x

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The average total cost (ATC) is the total cost per unit of output x

Hence, ATC = TC/x = (100+40x+2x^2)/x

ATC = (100/x) + 40 + 2x

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TC(x+1) = 100+40 (x+1) +2(x+1)^2

= 100 + 40x + 40 +2 (x^2 + 2x+ 1)

= 100 + 40x + 40 + 2x^2 + 4x + 2

Thus, TC(x+1) - TC(x) = 100 + 40x + 40 + 2x^2 + 4x + 2 - (100+40x+2x^2)

= 100 - 100 + 40x - 40x + 2x^2 - 2x^2 + 40 + 4x +2

= 40 + 4x + 2

= 42 + 4x

The expression: TC(x+1) - TC(x) means change in the total cost for 1 unit increase in the output produced.