Marginal Analysis Criterion for Maximal Average Profit
Closely related to Marginal Analysis for Minimum Average
Cost . Not quite the same as Marginal Analysis Criterion
for Maximum Profit . Hoffmann/Bradley, p. 242
Let P(q) is the total profit of producing the first q units.
Here is the graph of a typical P(q).
I won't tell you a specific formula for P(q). I will assume:
·
The graph of P(q) is smooth and concave downward.
·
P(q) = 0 has exactly two positive roots, the
smallest is called the break even point .
The average profit per unit, AvgP(q), of producing the
first q units, is
AvgP(q)
= P(q)/q.
Marginal Analysis Criterion for Maximal Average
Profit. Average profit per unit is maximized at the level of production where the average profit per unit equals the marginal profit; that is
AvgP(q) =
dP
dq
.
Th proof is the quotient rule for differentiation of P(q)/q.
Here is a graphical explanation of this criterion:
The average profit per unit at q is the slope of the
line from the origin 0 to the point (q, P(q)).
Look at the graph for various values of q,
Use a straight edge
or ruler to represent these lines.
As you move q to the right, the slope of the line from 0 to
(q, P(q)) increases and then decreases. The maximum slope occurs
when q » 2. At q » 2, the line from 0 to (q, P(q)) is tangent to the graph at (q, P(q)).
Here is an animated picture:
Note that the condition
P(q)
q
=
dP
dq
is
the same as
1 =
q
P
dP
dq
.
The quantity PE = [q/P] [dP/dq] is the
elasticity of profit with respect to output or output elasticity of
profit .
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version 4.00. On 03 Sep 2011, 21:26.