Demand, Revenue, Cost, and Profit Functions

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Demand, Revenue, Cost, and Profit Functions

• Math 174, Spring 2004

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Demand Function

• D(q) is the unit price at which q units of a
  product can be sold
• Experience suggest a quadratic demand function
• For your data, you will need to come up with a
  function based on test market data
• How should you do this?

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Units of Demand Function

• Will use two variables (not used in text)
• qu is the quantity in single units
• qk is the quantity in thousand units
• Test market data must be scaled up to the
  national level
• At national level, demand is presented as the
  unit price at which qk thousand drives can be
  sold

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Units of Demand Function

• We have the following proportion
• Note The size of your national market (in your
  data) is given in millions. You will have to
  rewrite this number in thousands so the units are
  correct.
• You can then solve the proportion for the
  national sales (in Ks)
• Round answers to nearest whole unit

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Demand Curve

• Once you have the national sales (in Ks) for
  each test market --
• Plot the points (qk vs. unit price)
• Fit a trend line to the data
• Show equation (make sure all coefficients go out
  to 8 decimal places)
• Use equation to determine points for demand curve
  need a lot of points
• Use points to plot a smooth demand curve

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Revenue Function

• Revenue, R(qk), is given in millions of dollars
  for selling qk thousand units
• So, R(qk) D(qk)qk 1000
• This answer is in millions
• For example, if D(qk)119.95 and qk 893, then
  R(qk)107,115,350
• We want the unit to be millions and also the
  answer to be a smaller number (so we can work
  with it easier)
• We can divide our answer by 1,000,000 to get
  about 107 million dollars
• So, our formula should be R(qk)(D(qk)qk)/1000

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Revenue Curve

• You need to use the data you generated for your
  demand curve and determine what the corresponding
  values are for your revenue function
• Use R(qk) function determined on previous slide
• When you graph your revenue data, you will have
  quantity (in Ks) along x-axis and total revenue
  (in Ms) along y-axis

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Cost Function

• If your total revenue is in millions of dollars,
  then your total cost must be also
• Total cost is the fixed cost plus the variable
  cost
• Fixed cost is in millions of dollars need to
  rewrite it so the unit is millions
• Example 21,600,000 is rewritten as 21.6 million
• You need to determine the variable cost for each
  amount of product you are producing
• We are going to define Marginal Cost to be the
  price, in dollars per item, for production at a
  given number of items

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Variable Cost

• To determine the variable cost, you will need to
  use the marginal cost of production
• For example, if you produce 1,200 drives
• The first 500 thousand cost 115 per drive
• (5001000115)/1,000,000 (500115)/100057.5
  million
• The next 600 thousand cost 100 per drive
• (600100)/1000 60 million
• The rest cost 90 per drive
• (10090)/1000 9 million
• The sum of the results give the final variable
  cost of 126.5 million

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COST function

• To determine the total cost, you can add the
  fixed cost and the variable cost
• To do this for every quantity, it would be
  tedious to determine a formula
• Can use COST function located under USER DEFINED
  formulas built in
• If you are generating your data and graphs in a
  new Excel workbook, you will need to copy the
  COST module over from Marketing Focus.xls
• Follow directions in slides 58-60 in Notebook or
  on CD

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Total Cost

• Regardless of how you calculate the total cost,
  you will need to understand each step
• Using the COST function make sure you
  understand why you are entering what you are for
  each of the arguments in the dialog box
• You should have a column that has the total cost
  for each particular quantity
• Once you have the data, you should create a graph
  that illustrates the cost display both the
  revenue and cost on one set of axes
• Why should you graph both curves together?
• Cost graph is quantity (Ks) vs. Total Cost (Ms)

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Profit

• Since the total cost and the total revenue is in
  millions, profit will also be in millions
• Profit Curve will be quantity (Ks) vs. total
  profit (Ms)
• Create a curve for profit by subtracting total
  revenue and total cost

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What should you do?

• Start a new Excel file, generate the data for
  your information
• Do not forget to copy the COST module
• Create each one of the curves talked about
  demand, revenue, cost, and profit
• Work with the same units and coefficient
  precision as is in the Class Project
• I will be solving each of your projects and will
  assume the same accuracy as the on-line text
• Your information should match mine at the end of
  the project
• Do not put this off! Your work will pile up if
  you leave too long
• It will also be a part of the next homework
  assignment