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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
  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)
 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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