Marginal Analysis Criterion for Minimal Average Cost
Hoffmann/Bradley, p. 243
C(q) is the total cost of producing the first q units.
The average cost, A(q), of producing the first q units, is
A(q)
= C(q)/q.
Marginal Analysis Criterion for Minimal Average Cost. Average cost is minimized at the level of production where average cost equals marginal cost; that is A(q) = C′(q). Here is a graphical explanation of this criterion:
For a typical C(q) [Frank/Bernanke, pp. 12 ff], the Marginal Cost,
[dC/dq], is increasing, or the graph of C(q) is concave
up!
Here is the graph of a typical C(q):
The average cost, A(q), of producing the first q units, is
A(q)
=
C(q)
q
.
The average cost, A(q), may be interpreted as the slope of the
line through (0,0), and (q, C(q)).
The (moving) box represents the point on the cost curve, the slope
of the (moving) red line represents the average cost.
Notice that the slope of the red line is minimized when the line
is tangent to the graph of C(q); i.e. A(q) = dC/dq.
Note that the condition
C(q)
q
=
dC
dq
is
the same as
1 =
q
C
dC
dq
.
The quantity CE = [q/C] [dC/dq] is the
elasticity of cost with respect to output or output-elasticity of
cost .
Then
percentagechangeinTotalCost C ≈ CE ·percentagechangeinoutput q.
When 0 < CE < 1 (relatively inelastic), A(q) is decreasing;
when 1 < CE (relatively elastic), A(q) is increasing.
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