Math 165: Optimizing Average Profit
Math 165: Optimizing Average Profit
PDF Version for Printing: 165.pdf


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).

./165avgprofit1.gif

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,

165avgprofit2.gif

Use a straight edge or ruler to represent these lines.

./maple/165avgprofit3.gif

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:

./maple/165avgprofit5animate.gif

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 .



File translated from TEX by TTHgold, version 4.00.
On 03 Sep 2011, 21:26.