# Profit, Cost, and Revenue

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```Profit, Cost, and Revenue
October 28, 2013
Profit, Cost, and Revenue
Maximizing profit
A fundamental issue for a producer is how to maximizing
profit.
Profit, Cost, and Revenue
Maximizing profit
A fundamental issue for a producer is how to maximizing
profit.
π(q) = R(q) − C (q)
Profit, Cost, and Revenue
Maximizing profit
A fundamental issue for a producer is how to maximizing
profit.
π(q) = R(q) − C (q)
MC = C 0 : Marginal cost; MR = R 0 : Marginal revenue
Profit, Cost, and Revenue
Example
Estimating maximum profit if the revenue and cost are given by
the curves R and C , respectively, in the figure.
C
\$ (thousands)
R
150
95
20
40
60
80
100
q (quantity)
Profit, Cost, and Revenue
Example
Profit =Revenue − Cost
C
\$ (thousands)
R
150
95
20
40
60
80
100
q (quantity)
Profit, Cost, and Revenue
Example
Profit =Revenue − Cost
Profit is represented by the the vertical distance from curve C
to curve R, marked by vertical arrow.
C
\$ (thousands)
R
150
95
20
40
60
80
100
q (quantity)
Profit, Cost, and Revenue
Example
Profit =Revenue − Cost
Profit is represented by the the vertical distance from curve C
to curve R, marked by vertical arrow.
Arrow is going down =⇒ No profit
C
\$ (thousands)
R
150
95
20
40
60
80
100
q (quantity)
Profit, Cost, and Revenue
Example
Profit =Revenue − Cost
Profit is represented by the the vertical distance from curve C
to curve R, marked by vertical arrow.
Arrow is going down =⇒ No profit
Arrow is going up =⇒ Making profit
C
\$ (thousands)
R
150
95
20
40
60
80
100
q (quantity)
Profit, Cost, and Revenue
Example
Profit =Revenue − Cost
Profit is represented by the the vertical distance from curve C
to curve R, marked by vertical arrow.
Arrow is going down =⇒ No profit
Arrow is going up =⇒ Making profit
Profit is maximized if the arrow is going up and has the
largest distance.
C
\$ (thousands)
R
150
95
20
40
60
80
100
q (quantity)
Profit, Cost, and Revenue
Example
We now analyze the marginal costs and marginal revenues
near the optimal point.
MC<MR
MC>MR
R
C
MC=MR
Profit, Cost, and Revenue
Example
We now analyze the marginal costs and marginal revenues
near the optimal point.
Global maxima and minima occur at critical points or at the
endpoints of the interval
MC<MR
MC>MR
R
C
MC=MR
Profit, Cost, and Revenue
Example
We now analyze the marginal costs and marginal revenues
near the optimal point.
Global maxima and minima occur at critical points or at the
endpoints of the interval
π 0 (q) = R 0 (q) − C 0 (q) = 0
MC<MR
MC>MR
R
C
MC=MR
Profit, Cost, and Revenue
Example
We now analyze the marginal costs and marginal revenues
near the optimal point.
Global maxima and minima occur at critical points or at the
endpoints of the interval
π 0 (q) = R 0 (q) − C 0 (q) = 0
MR = R 0 = C 0 = MC .
MC<MR
MC>MR
R
C
MC=MR
Profit, Cost, and Revenue
Conclusion
The maximum or minimum profit can occur where marginal
profit=0. That is where marginal revenue=marginal cost.
Profit, Cost, and Revenue
Example
The (total) revenue and (total) cost curves for a product are given
in the Figure.
(a) Sketch (roughly) the marginal cost and revenue.
(b) Graph the profit function π(q).
Profit, Cost, and Revenue
Example
Profit, Cost, and Revenue
Example
Find the quantity which maximizes the profit if the total revenue
and total cost (in dollars) are given by
R(q) = 5q − 0.003q 2
C (q) = 300 + 1.1q
where q is quantity and 0 ≤ q ≤ 1000 units. What production
level gives the maximize profit?
Profit, Cost, and Revenue
Maximize Revenue
At a price of \$80 for a half-day trip, a white-water rafting company
attracts 300 customers. Every \$5 decrease in price attracts an
Find the demand equation.
Profit, Cost, and Revenue
Maximize Revenue
At a price of \$80 for a half-day trip, a white-water rafting company
attracts 300 customers. Every \$5 decrease in price attracts an
Find the demand equation.
Express revenue as a function of price
Profit, Cost, and Revenue
Maximize Revenue
At a price of \$80 for a half-day trip, a white-water rafting company
attracts 300 customers. Every \$5 decrease in price attracts an
Find the demand equation.
Express revenue as a function of price
What price should the company charge per trip to maximize
revenue?
Profit, Cost, and Revenue
Maximize Revenue
At a price of \$80 for a half-day trip, a white-water rafting company
attracts 300 customers. Every \$5 decrease in price attracts an