#### Transcript Demand, Revenue, Cost, & Profit

```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]
= 9000+177*q0.633
• 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
\$14,000.00
2,000
\$22,000.00
3,000
\$28,000.00
Variable cost function
• Assume that we are going to fit a power
function
• VC(q) = u * qv (here u and v are constants)
Variable Costs Function
0.633
VC(q)
y = 177x
\$50,000
\$40,000
\$30,000
\$20,000
\$10,000
\$0
0
1,000
2,000
q
3,000
4,000
Cost function
Recall
q
C(q) = C0 + VC(q).
= 9000+177*q0.633
C(q)
Cost Function
\$50,000
\$40,000
\$30,000
\$20,000
\$10,000
\$0
0
1000
2000
q
3000
4000
C(q)
0
\$9,000.00
8
\$9,660.13
16
\$10,023.72
24
\$10,323.27
32
\$10,587.57
40
\$10,828.43
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)
Profit=Revenue-Cost
Sample Data Points
q
C(q)
R(q)
P(q)
0
\$9,000.00
\$0.00
-\$9,000.00
8
\$9,660.13
\$241.50
-\$9,418.63
16
\$10,023.72
\$482.96
-\$9,540.76
24
\$10,323.27
\$724.37
-\$9,598.90
32
\$10,587.57
\$965.72
-\$9,621.85
40
\$10,828.43
\$1,207.01
-\$9,621.41
Profit Function-Dinner problem
P(q)
Profit Function
\$15,000
\$10,000
\$5,000
\$0
-\$5,000 0
-\$10,000
1000
2000
q
3000
4000
Summary –Dinner Problem
Cost
Revenue
Revenue and Cost Function
\$50,000
\$30,000
\$20,000
\$10,000
\$0
0
1000
2000
q
3000
4000
Profit Function
\$15,000
\$10,000
P(q)
Dollars
\$40,000
\$5,000
\$0
-\$5,000
0
1000
2000
-\$10,000
q
3000
4000
Project Focus
• How can demand, revenue,cost, and profit
functions help us price T/2 Mega 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
• 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
Rough estimates based on
Graphs of D(q), P(q)
400
800
1,200
1,600
2,000
P (q ) (M's)
q (K's)
Demand Data
Price
• Optimal Quantity1200
• Optimal Price\$300
• Optimal Profit\$42M
Profit Function
\$70
\$60
\$50
\$40
\$30
\$20
\$10
\$0
-\$10 0
-\$20
\$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)
Goals
•
1. What price should Storage 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 T/2 Mega?
•
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?
32
Goals-Contd.
•
6. What profit could Storage Tech expect, if they price the
drives at \$299.99?
•
7. How much should Storage Tech pay for an advertising
campaign that would increase demand for the T/2 Mega 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 Storage Tech to put \$15,000,000
into training and streamlining which would reduce the variable
production costs by 7% for the coming year?
33
What’s next?
• So far we have graphical estimates for
some of our project questions(Q1-3 only)
• We need now is some way to replace
graphical estimates with more precise
computations
```