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) = 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)
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
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
Variable cost function • Assume that we are going to fit a power function • VC(q) = u * qv (here u and v are constants)
Cost function Recall C(q) = C0 + VC(q). = 9000+177*q0.633
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)
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 • Show marketing data.xls (How to calculate)
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
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
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)
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-1200 • Optimal Price- $300 • Optimal Profit-$42M
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?
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?
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