Demand, Revenue, Cost,

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Demand, Revenue, Cost, Profit
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Demand Function D(q)

• p D(q)
• In this function the input is q and output p
• q-independent variable/p-dependent variable
• Recall yf(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?

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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

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Example

Define the demand function to be D(q) a?q2
b?q c, where a ?0.0000018, b ?0.0002953,
and c 30.19.
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Example problem( Dinner.xls)

• Restaurant wants to introduce a new buffalo steak
  dinner
• Test prices (Note these are unit prices)
• If I want the demand function, what is our
  input/output?
• Recall pD(q)

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Revenue Function R(q)

• R(q)qD(q)
• The amount that a producer receives from the sale
  of q units
• Recall pD(q)
• What is p?
• -unit price per item
• Revenue number of unitsunit price

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Example
Sample Data Points Sample Data Points 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
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Cost Function

• A producers 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.
• Recallfixed cost do not depend upon the amount
  of a good that is produced

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Example
Fixed Cost Fixed Cost
C0 9,000.00

Variable Costs Variable Costs
Number of Dinners(q) Cost-VC(q)
1,000 21,000.00
2,000 30,000.00
3,000 36,000.00
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D, R, C, P, Expenses Profit
Note that VC and C are only plotted over the
intervals where the logarithmic model is believed
to apply.
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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)

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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)

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D, R, C, P, Expenses Profit
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Project Focus

• How can demand, revenue,cost, and profit
  functions help us price 12-GB drives?
• Must find the demand, revenue and cost functions

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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.

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Projected yearly sales -National market

• We have the information about the Test markets
  Potential national market size
• Show marketing data.xls (How to calculate)

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Demand function-Project1D(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

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Demand Function (continued)

• D(q) -0.00005349q2 -0.03440302q 414.53444491

Marketing Project
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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

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Revenue function-R(q)

• Revenue in dollars D(q)q1000
• Revenue in millions of dollars
  D(q)q1000/1000000
• D(q)q/1000
• Why do this conversion?
• Revenue should be in millions of dollars

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Revenue function
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Total cost function-C(q)

• C(q)-Cost, in millions of dollars,of producing q
  thousand drives

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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

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Cost Function

• The cost function, C(q), gives the relationship
  between total cost and quantity produced.
• User defined function COST in Excel.

Marketing Project
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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)

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Revenue Cost Functions
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Main Focus-Profit

• Recall P(q)-the profit, in millions of dollars
  from selling q thousand drives
• P(q)R(q)-C(q)

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Profit Function

• The profit function, P(q), gives the relationship
  between the profit and quantity produced and
  sold.
• P(q) R(q) C(q)

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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?

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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?

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Whats 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