AVERAGE COST ANALYSIS Below you see a model of the cost (TOTAL COST) of a certain company to produce x units of a certain product

C( x ) := 900 + 3 ⋅ x + x

2

C( 10) = 1030 euro

i.e.: If you produce 10 units the total cost is:

C( 40) = 2620

If you produce 40 units the total cost is:

The derivative of the cost is:

euro

d C( x ) → 3 + 2⋅ x dx

For all values in the domain the derivative is positive; which means that the cost function is always increasing in its domain. Lets verify this with the graph

6000 4000 C( x) 2000

0

20

40

60

x

The average cost per unit (in short Average cost), is defined as the ration of Total cost withe the number of units produced. So:

Average Cost per unit ---->

AC( x ) :=

C( x ) x

If we substitute in the function of cost and divide the x into every term of the numerator we get the function: AC( x ) :=

AC( x ) :=

(900 + 3⋅x + x2) x 900 x

+3+x

so :

The domain of the Average cost is ( 0 , + ∞ ) Lets compute the limits at the end points to see what how the function is behaving at those end points lim

x → 0+

lim

AC( x ) → ∞

AC( x ) → ∞

x→∞

The above limits tell us the following: • for very small production levels the average cost per unit is very large • for very large production levels the average cost is again very large Lets take a look:

400

AC ( x) 200

0

50 x

What do we see above: • The average cost per unit starts high and at the beginning drops really fast. This is because we have the Fixed Costs to spread in the beginning • at some point the cost per unit becomes minimum • after that the cost per unit slowwly rises (think that as we need to produce more we might have extra cost for maitenance of production machines, overtime, extra workers on the production line, ...) To find the minimum value, we need to find the horizontal tangent point. That is locate the x value for which the derivative of the average cost is zero. I will use a Given/Find to locate this x-value Given d AC( x ) = 0 dx Find( x ) → ( 30 −30 ) The minimum Average cost per unit is: AC( 30) = 63 The Average cost pre unit is minimized when we produce 30 units and this minimum value is 63 euro per unit

Relation of Marginal Cost (MC) with Average Co st (AC) The marginal cost function is the derivative of the total cost function, below:

MC( x ) :=

d C( x ) → 3 + 2⋅ x dx

Lets look at the graph below:

300 AC ( x)

200

MC ( x) AC ( 30) 100

0

20

40

60

x , x , 30

Notice that the Marginal Cost and the Average Cost meet at point where the Average cost is minimized. Coincidence?

NO ! It turns out that this is THEORY. AN other way to locate the minimum Average Cost (besides locating the root of the derivative of the Average Cost), is to find when the Marginal Cost (the derivative of Total Cost) is equal to the Average Cost Lets have the computer solve this equation below Given

MC( x ) = AC( x )

Find( x ) → ( 30 −30 )

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