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 Define the demand function to be D(q) = a  q 2 + b  q + c, where a =  , b =  , and c =

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)

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 Sample Data Points qD(q)D(q)R(q)R(q) 0$30.19$0.00 8$30.19$ $30.18$ $30.18$ $30.18$ $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) = C 0 + VC(q) =fixed cost + variable cost [here VC(q)-variable cost for q units of a good] = *q Recall:fixed cost do not depend upon the amount of a good that is produced

Example Fixed Cost C0C0 $9, Variable Costs Number of Dinners(q)Cost-VC(q) 1,000$14, ,000$22, ,000$28,000.00

Variable cost function Assume that we are going to fit a power function VC(q) = u * q v (here u and v are constants)

Cost function Recall C(q) = C 0 + VC(q). = *q qC(q)C(q) 0$9, $9, $10, $10, $10, $10,828.43

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 qC(q)C(q)R(q)R(q)P(q)P(q) 0$9,000.00$0.00-$9, $9,660.13$ $9, $10,023.72$ $9, $10,323.27$ $9, $10,587.57$ $9, $10,828.43$1, $9,621.41

Profit Function-Dinner problem

Summary –Dinner Problem

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)

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) D(q) = q q 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/ = D(q)*q/1000 Why do this conversion? Revenue should be in millions of dollars

Revenue function

Total cost function-C(q) C(q)-Cost, in millions of dollars,of producing q thousand drives

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. 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

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)

Rough estimates based on Graphs of D(q), P(q) Optimal Quantity Optimal Price- $300 Optimal Profit- $42M

32 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?

33 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(Q1-3 only) We need now is some way to replace graphical estimates with more precise computations