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

ExampleDemand Function

y = -0.0000018x2 - 0.0002953x + 30.19

$0

$8

$16

$24

$32

0 1,000 2,000 3,000 4,000q

D(q

)

Define the demand function to be D(q) = aq2 + bq + 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)

• If I want the demand function, what is our input/output?

• Recall p=D(q)

Price $14.95 $19.95 $24.95 $29.95Number sold per week 2,800 2,300 1,600 300

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

ExampleRevenue Function

$0

$10,000

$20,000

$30,000

$40,000

$50,000

0 1000 2000 3000 4000q

R(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]

. Hence, they assume that there are constants u and v such that VC(q) = uln(q) + v, over a range of values for q between 1,000 and 4,000.

• 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 $21,000.00

2,000 $30,000.00

3,000 $36,000.00

D, R, C, & P, Expenses & Profit Variable Costs Function

y = 13581.51Ln(x) - 72929.37

$0

$10,000

$20,000

$30,000

$40,000

$50,000

0 1,000 2,000 3,000 4,000q

VC

(q)

Cost Function

$0

$10,000

$20,000

$30,000

$40,000

$50,000

0 1000 2000 3000 4000q

C(q

)

Note that VC and C are only plotted over the intervals where the logarithmic model is believed to apply.

Cost function

• The total weekly cost function, over that range, for the buffalo steak dinners is

• C(q) = C0 + VC(q) = 9,000 + 13,581.51ln(q) 72,929.37 = 63,929.37 + 13,581.51ln(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)

D, R, C, & P, Expenses & Profit

Profit Function

-$6,000

-$4,000

-$2,000

$0

$2,000

$4,000

$6,000

0 1000 2000 3000 4000

q

P(q

)

Revenue and Cost Function

$0$10,000

$20,000$30,000$40,000

$50,000$60,000

0 1000 2000 3000 4000q

Dol

lars

Revenue

Cost

Project Focus

• How can demand, revenue,cost, and profit functions help us price 12-GB 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

• Show marketing data.xls (How to calculate)

)'(]1[

]1[1)'( sKmarketnationalofsize

markettestofsize

salesmarkettestmarkettestforsKsalesnational

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

Demand Function (continued)

D(q) =-0.00005349q2 + -0.03440302q + 414.53444491

Marketing Project

Demand Data

y = -0.00005349x2 - 0.03440302x + 414.53444491

$0$100$200$300$400$500

0 400 800 1,200 1,600 2,000 2,400 2,800

Quantity (K's)

Pri

ce

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 functionRevenue Function

$0

$100

$200

$300

$400

$500

0 400 800 1,200 1,600 2,000 2,400 2,800

q (K's)

R(q

) (M

's)

Total cost function-C(q)

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

Fixed Cost (M's)

$135.0 Marginal Cost1 First 800 $160.002 Second 400 $128.003 Further $72.00

Variable Costs (M's)

Batch Size (K's)

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.

User defined function COST in Excel.

2001 if0001

2001722314

2001800 if0001

800128263

8000 if0001

160135

,q,

),q(.

,q,

)q(

q,

q

)q(C

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

$0

$100

$200

$300

$400

$500

0 400 800 1,200 1,600 2,000 2,400 2,800

q (K's)

(M's

)

Revenue

Cost

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

-$20-$10

$0$10$20$30$40$50$60$70

0 400 800 1,200 1,600 2,000

q (K's)

P(q

) (M

's)

29

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?

30

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?

What’s 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