Demand, Revenue, Cost, &
Profit
Demand Function – D(q)
• p =D(q)
• In this function the input is q and output p
• q-independent variable/p-dependent variable
[Recall y=f(x)]
• p =D(q) the price at which q units of the good can be
sold
• Unit price-p
• Most demand functions- Quadratic [ PROJECT 1]
• Demand curve, which is the graph of D(q), is generally
downward sloping
– Why?
Demand Function – D(q)
• As quantity goes down, what happens to
price?
-price per unit increases
• As quantity goes up, what happens to
price?
-price per unit decreases
Example
Demand Function
y = -0.0000018x2 - 0.0002953x + 30.19
$32
D(q)
$24
$16
$8
$0
0
1,000
2,000
q
3,000
Define the demand function to be
D(q) = aq2 + bq + c, where a = 0.0000018,
b = 0.0002953, and c = 30.19.
4,000
Example problem( Dinner.xls)
• Restaurant wants to introduce a new buffalo
steak dinner
• Test prices (Note these are unit prices)
Price
$14.95 $19.95 $24.95 $29.95
Number sold per week 2,800 2,300 1,600
300
• If I want the demand function, what is our
input/output?
• Recall p=D(q)
Revenue Function – R(q)
• R(q)=q*D(q)
• The amount that a producer receives from
the sale of q units
• Recall p=D(q)
• What is p?
-unit price per item
• Revenue= number of units*unit price
Example
Revenue Function
$50,000
R(q)
$40,000
$30,000
$20,000
$10,000
$0
0
1000
2000
q
3000
Sample Data Points
q
D(q)
R(q)
0
$30.19
$0.00
8
$30.19
$241.50
16
$30.18
$482.96
24
$30.18
$724.37
32
$30.18
$965.72
40
$30.18
$1,207.01
4000
Cost Function
A producer’s total cost function, C(q), for the production of q units is given
by
C(q) = C0 + VC(q)
=fixed cost + variable cost
[here VC(q)-variable cost for q units of a good]
.
Hence, they assume that there are constants u and v such that
VC(q) = uln(q) + v, over a range of values for q between 1,000 and
4,000.
• Recall:fixed cost do not depend upon the
amount of a good that is produced
Example
Fixed Cost
C0
$9,000.00
Variable Costs
Number of Dinners(q)
Cost-VC(q)
1,000
$21,000.00
2,000
$30,000.00
3,000
$36,000.00
D, R, C, & P, Expenses & Profit
Variable Costs Function
$50,000
y = 13581.51Ln(x) - 72929.37
$40,000
VC(q)
Note that VC
and C are only plotted
over the intervals where
the logarithmic model is
believed to apply.
$30,000
$20,000
$10,000
$0
0
Cost Function
$50,000
C(q)
$40,000
$30,000
$20,000
$10,000
$0
0
1000
2000
q
3000
4000
1,000
2,000
q
3,000
4,000
Cost function
• The total weekly cost function, over that range, for the
buffalo steak dinners is
• C(q) = C0 + VC(q) = 9,000 + 13,581.51ln(q)  72,929.37
= 63,929.37 + 13,581.51ln(q)
Profit Function
• let P(q) be the profit obtained from
producing and selling q units of a good
at the price D(q).
• Profit = Revenue  Cost
• P(q) = R(q)  C(q)
D, R, C, & P, Expenses & Profit
Revenue
Revenue and Cost Function
$60,000
$50,000
$40,000
$30,000
$20,000
$10,000
$0
0
1000
2000
q
3000
4000
Profit Function
$6,000
$4,000
$2,000
P(q)
Dollars
Cost
$0
-$2,000 0
1000
2000
-$4,000
-$6,000
q
3000
4000
Project Focus
• How can demand, revenue,cost, and profit
functions help us price 12-GB drives?
• Must find the demand, revenue and cost
functions
Important – Conventions for units
•  Prices for individual drives are given in
dollars.
•
 Revenues from sales in the national
market are given in millions of dollars.
•
 Quantities of drives in the test
markets are actual numbers of drives.
•
 Quantities of drives in the national
market are given in thousands of drives.
Projected yearly sales –
-National market
• We have the information about the Test markets
& Potential national market size
national sales ( K ' s) for test market 1 
[test market 1 sales ]
 size of national market( K ' s )
[ size of test market 1]
• Show marketing data.xls (How to calculate)
Demand function-Project1
D(q)
• D(q) –gives the price, in dollars per drive
at q thousand drives
• Assumption – Demand function is
Quadratic
• The data points for national sales are
plotted and fitted with a second degree
polynomial trend line
• Coefficients- 8 decimal places
Demand Function (continued)
Price
Demand Data
$500
$400
$300
$200
$100
$0
2
y = -0.00005349x - 0.03440302x + 414.53444491
0
400
800
1,200 1,600 2,000 2,400 2,800
Quantity (K's)
D(q) =-0.00005349q2 + -0.03440302q + 414.53444491
Marketing Project
Revenue function- Project1
R(q)
• R(q) is to give the revenue, in millions of
dollars from selling q thousand drives
• Recall D(q)- gives the price, in dollars per
drive at q thousand drives
• Recall q – quantities of drives in the
national market are given in thousand of
drives
Revenue function-R(q)
• Revenue in dollars= D(q)*q*1000
• Revenue in millions of dollars = D(q)*q*1000/1000000
= D(q)*q/1000
• Why do this conversion?
Revenue should be in millions of dollars
Revenue function
Revenue Function
$500
R (q ) (M's)
$400
$300
$200
$100
$0
0
400
800
1,200
1,600
q (K's)
2,000
2,400
2,800
Total cost function-C(q)
• C(q)-Cost, in millions of dollars,of producing q
thousand drives
Fixed Cost
(M's)
$135.0
Variable Costs (M's)
1
2
3
Batch Size (K's)
First
800
Second
400
Further
Marginal Cost
$160.00
$128.00
$72.00
Total cost function-C(q)
• Depends upon 7 numbers
– q(quantity)
– Fixed cost
– Batch size 1
– Batch size 2
– Marginal cost 1
– Marginal cost 2
– Marginal cost 3
Cost Function
 The cost function, C(q), gives the relationship
between total cost and quantity produced.
160q

135

if 0  q  800

1,000

128( q  800 )

C( q )  263 
if 800  q  1,200
1,000


314.2  72( q  1,200 ) if q  1,200

1,000
 User defined function COST in Excel.
Marketing Project
How to do the C(q) in Excel
• We are going to use the COST
function(user defined function)
• All teams must transfer the cost function
from Marketing Focus.xls to their project1
excel file
• Importing the COST function(see class
webpage)
Revenue & Cost Functions
Revenue & Cost Functions
$500
(M's)
$400
Revenue
$300
Cost
$200
$100
$0
0
400
800
1,200
1,600
q (K's)
2,000
2,400
2,800
Main Focus-Profit
• Recall P(q)-the profit, in millions of dollars
from selling q thousand drives
• P(q)=R(q)-C(q)
Profit Function
 The profit function, P(q), gives the relationship
between the profit and quantity produced and sold.
 P(q) = R(q) – C(q)
P (q ) (M's)
Profit Function
$70
$60
$50
$40
$30
$20
$10
$0
-$10 0
-$20
400
800
1,200
q (K's)
1,600
2,000
Goals
•
1. What price should Card Tech put on the drives,
in order to achieve the maximum profit?
•
2. How many drives might they expect to sell at
the optimal price?
•
3. What maximum profit can be expected from
sales of the 12-GB?
•
4. How sensitive is profit to changes from the
optimal quantity of drives, as found in Question 2?
•
5. What is the consumer surplus if profit is
maximized?
29
Goals-Contd.
•
6. What profit could Card Tech expect, if they price the
drives at $299.99?
•
7. How much should Card Tech pay for an advertising
campaign that would increase demand for the 12-GB drives by 10%
at all price levels?
•
8. How would the 10% increase in demand effect the
optimal price of the drives?
•
9. Would it be wise for Card Tech to put $15,000,000 into
training and streamlining which would reduce the variable production
costs by 7% for the coming year?
30
What’s next?
• So far we have graphical estimates for
some of our project questions
• We need now is some way to replace
graphical estimates with more precise
computations