# Demand_ Revenue_ Cost_ _ Profit

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```					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
\$24
D(q)

\$16
\$8
\$0
0        1,000      2,000     3,000         4,000
q

Define the demand function to be
D(q) = aq2 + bq + c, where a = 0.0000018,
b = 0.0002953, and c = 30.19.
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
\$40,000
\$30,000
R(q)

\$20,000
\$10,000
\$0
0            1000        2000         3000         4000
q

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
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
y = 177x
\$50,000
\$40,000
VC(q)

\$30,000
\$20,000
\$10,000
\$0
0   1,000    2,000   3,000   4,000

q
Cost function
Recall         C(q) = C0 + VC(q).              q        C(q)
0    \$9,000.00
= 9000+177*q0.633               8    \$9,660.13
16   \$10,023.72
Cost Function                     24   \$10,323.27
32   \$10,587.57
\$50,000
\$40,000                                           40   \$10,828.43
C(q)

\$30,000
\$20,000
\$10,000
\$0
0      1000    2000   3000   4000

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)
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

Profit Function

\$15,000
\$10,000
\$5,000
P(q)

\$0
-\$5,000 0     1000     2000   3000   4000
-\$10,000

q
Summary –Dinner Problem
Revenue and Cost Function          Cost
Revenue
\$50,000
\$40,000
Dollars

\$30,000
\$20,000
\$10,000
\$0
0    1000     2000    3000      4000
q                                         Profit Function
\$15,000

\$10,000

\$5,000
P(q)

\$0
0    1000     2000    3000   4000
-\$5,000

-\$10,000
q
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

[test market 1 sales]
national sales( K ' s ) for test market 1                              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)
Demand Data

\$500
\$400                       2
y = -0.00005349x - 0.03440302x + 414.53444491
Price

\$300
\$200
\$100
\$0
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

\$400
R (q ) (M's)

\$300

\$200

\$100

\$0
0    400   800   1,200    1,600   2,000   2,400   2,800
q (K's)
Total cost function-C(q)
• C(q)-Cost, in millions of dollars,of producing q
thousand drives

Fixed Cost
Variable Costs (M's)
(M's)
\$135.0                   Batch Size (K's)    Marginal Cost
1         First             800     \$160.00
2        Second             400     \$128.00
3        Further                    \$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

\$400
Revenue
\$300
(M's)

Cost
\$200

\$100

\$0
0   400    800    1,200   1,600      2,000   2,400   2,800
q (K's)
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)

Profit Function
\$70
\$60
\$50
P (q ) (M's)

\$40
\$30
\$20
\$10
\$0
-\$10 0   400     800             1,200   1,600   2,000
-\$20
q (K's)
Rough estimates based on
Graphs of D(q), P(q)
• Optimal Quantity-
Profit Function
\$70
\$60
\$50

1200

P (q ) (M's)
\$40
\$30
\$20

• Optimal Price-
\$10
\$0
-\$10 0      400         800             1,200   1,600   2,000
-\$20
\$300                                                                     q (K's)

• Optimal Profit-      Demand Data

\$42M                        \$500
\$400                                         2
y = -0.00005349x - 0.03440302x + 414.53444491
Price

\$300
\$200
\$100
\$0
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

```
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 views: 4 posted: 7/20/2012 language: pages: 34