Pricing I: Linear Demand This module covers the relationships between price and quantity, maximum willing to buy, maximum reservation price, profit maximizing price, and price elasticity, assuming a linear relationship between price and demand. Authors: Paul Farris and Phil Pfeifer Marketing Metrics Reference: Chapter 7 2010-14 Paul Farris, Phil Pfeifer and Management by the Numbers, Inc.
Definition Linear Price-Quantity Functions Linear demand functions are those in which the relationship between quantity and price is linear. This means that any identical change in price (no matter what the starting price) produces an identical change in units demanded. The per unit change in Q caused by a change in P is called the slope. With linear demand curves, the slope is constant (the same for all prices). The demand curve is actually a demand line. LINEAR PRICE-QUANTITY FUNCTIONS This presentation covers the topics of maximum willing to buy (MWB), maximum reservation price (MRP), profit maximizing price, and price elasticity under the assumption of a linear relationship between price and quantity. MBTN Management by the Numbers 2
Quantity 10 9 8 7 6 5 4 3 2 1 0 Definition Linear Price-Quantity Demand Increasing price from $0.50 to $1.50 causes a drop in quantity from 9 to 7 units. Thus, the slope (delta Q / delta P) is 2 = (9 7) / (1.5.50). Increasing price from $2 to $3 causes a drop in quantity from 6 to 4. Thus, the slope (delta Q / delta P) is also -2, for each $1 change in price we see a change on 2 unit of quantity sold in the opposite direction. 0 $1 $2 $3 $4 $5 Price Slope of demand = change in quantity / change in price. Insight Price slope is almost always negative, but often people drop the sign. LINEAR PRICE-QUANTITY DEMAND 3
Linear Price-Quantity Demand Formula MWB is where P = 0 Quantity 10 9 8 7 6 5 4 3 2 1 0 MWB -- Maximum Willing to Buy Formulas for linear demand functions use the following format (recall y = mx + b): Quantity = Slope * Price + MWB In this example, beta = - 2. One could then solve for MWB and MRP using this equation and a price quantity point. 0 $1 $2 $3 $4 $5 Price MRP -- Maximum Reservation Price MRP is where Q = 0. LINEAR PRICE-QUANTITY DEMAND FORMULA MBTN Management by the Numbers 4
Definitions Linear Price-Quantity Demand Formula Maximum Willingness to Buy (MWB) = Quantity - Slope * Price Conceptually, MWB is the quantity that would be sold (presuming linear demand) if the price is free (price = 0). Maximum Reservation Price (MRP) = MWB / (- Slope) Conceptually, MRP is the maximum price where some quantity would be sold (again, presuming linear demand). This is actually where the linear demand line crosses the axis or where quantity = 0. LINEAR PRICE-QUANTITY DEMAND FORMULA We took a bit of a leap of faith there. For some of you, it may be helpful to review the how these formulas were derived which we ll do on the following slide. MBTN Management by the Numbers 5
Linear Price-Quantity Demand Formula You may recall from algebra that linear functions follow the format: y = mx + b Where b is the Y intercept (where the line crosses the y axis or where x = 0), m is the slope of the line, and x and y are the coordinates of any point on the line. First, let s express the basic linear function using marketing terminology: Quantity = Slope * Price + MWB We can use this basic equation for price / quantity relationships as well as bringing in some other important managerial concepts such as the maximum willingness to buy (MWB) and maximum reservation price (MRP). With a little substitution, the equation is, Quantity = MWB * [1 Price / MRP] From this, we can derive functions solving for MWB and MRP in the definitions. LINEAR PRICE-QUANTITY DEMAND FORMULA Definitions Maximum Willingness to Buy (MWB) Maximum Reservation Price (MRP) = Quantity - Slope * Price = MWB / (- Slope) MBTN Management by the Numbers 6
10 9 8 7 Quantity 6 5 4 3 2 1 0 Optimal (Profit Maximizing) Price MWB Now let s turn to the question of maximizing profit using the following example. If you are told that the unit cost is $1, what is the optimal (profitmaximizing) price to charge? 0 $1 $2 $3 $4 $5 Price MRP OPTIMAL (PROFIT MAXIMIZING) PRICE Unit Cost MBTN Management by the Numbers 7
Answer Quantity 10 9 8 7 6 5 4 3 2 1 0 Optimal (Profit Maximizing) Price MWB As you might expect, the profit maximizing price is always more than cost and less than the MRP. Profit = Q * (P-C) = 4 * ($3-$1) = $8 0 $1 $2 $3 $4 $5 Price But, it is nice, and maybe a little surprising, that the profit-maximizing price is ALWAYS exactly half-way between unit costs and MRP. MRP OPTIMAL (PROFIT MAXIMIZING) PRICE C = Unit Cost, P = Price, Q = Quantity MBTN Management by the Numbers 8
Big Conclusion! So, for linear demand functions, we only need two pieces of information to calculate the profit maximizing price: 1. Unit Variable Cost and 2. Maximum Reservation Price (MRP is where Q is equal to 0) BIG CONCLUSION! Definition Profit Maximizing Price = ½ (Unit Cost + MRP) In addition, since linear demand functions have a constant slope, with any two points (price quantity combinations) you can calculate the slope, the MWB, and the MRP. Now let s apply these definitions in a couple of examples. Question 1: For a product, we observe that at a price of $5, a quantity of 8 is sold and that at $4 we sell 12 units of a product. The cost of the product is $3. What is the MWB, MRP and Profit Maximizing Price? MBTN Management by the Numbers 9
Example - How to Calculate MWB & MRP First, we have two data points, necessary to calculate the slope which we ll need for solving for MWB. Slope = delta Q / delta P = (8 12) / (5 4) = 4 Using the MWB = Quantity - Slope * Price formula, substitute one set of price & quantity values from above along with the slope. MWB = 12 (-4 * 4) MWB = 12 (-16) = 12 + 16 = 28 Verify with the other point: MWB = 8 (-4 * 5) = 8 + 20 = 28 (Matches) Now, it is easy to find MRP by solving for the price at which we sell zero units. 0 = 28 4 * price = $7 or by using MRP = MWB / (- Slope) MRP = 28 / - (-4) = 28 / 4 = $7 EXAMPLE OF HOW TO CALCULATE MWB AND MRP Profit Maximizing Price = 1/2 (Cost + MRP) = ½ (3 + 7) = $5. Answers: MBW = 28, MRP = $7, Profit Maximizing Price = $5 MBTN Management by the Numbers 10
Finding Optimal Price with Regression Question 2: Suppose we apply regression to sales data, and find the following demand function where slope = 4 and MWB = 100: q = slope * p + MWB = 4 * p + 100 If the unit cost equals $5, what is the optimal price? Answer: First, find MRP by dividing MWB by slope (e.g. a + value), 100 / 4 = 25, so MRP = 25 Then add to the cost ½ the difference between cost and MRP. Cost + ½ (MRP Cost) = $5 + ½ ($25 $5) = $15 So, the profit-maximizing price is $15. FINDING OPTIMAL PRICE WITH REGRESSION MBTN Management by the Numbers 11
Slope versus Elasticity The slope is the unit change in quantity for a small unit change in price. For linear demand curves, the slope is constant (the same at all prices). Another measure of how much quantity reacts to changing prices is ELASTICITY. Whereas slope is a unit per unit change rate, elasticity is a percentage per percentage change rate. It is the slope times (P/Q), and is often thought of as the percentage change in Q for a small percentage change in P. SLOPE VERSUS ELASTICITY MBTN Management by the Numbers 12
Slope versus Elasticity For a linear demand curve, the slope is a constant. This means that the elasticity will NOT be a constant but will depend on the initial price. For any linear demand curve, the elasticity is larger for higher prices. This makes sense because if a unit change in price produces a constant unit change in Q, the unit increase in price is a smaller percentage of P if P is high and the unit decrease in Q is a larger percentage of Q when Q is low (which it is if P is high). SLOPE VERSUS ELASTICITY MBTN Management by the Numbers 13
Q 10 9 8 7 6 5 4 3 2 1 0 Linear Price-Quantity Demand Increasing price from $0.50 to $1.50 causes a drop in quantity from 9 to 7 units. Thus, the slope (delta Q / delta P) is 2 and the elasticity ( 2 / 9) / ($1 / $0.5) =.11 Increasing price from $2 to $3 causes a drop in quantity from 6 to 4. Thus, the slope (delta Q/delta P) is 2, but the elasticity is ( 2 / 6) / ($1 / $2) =.67 Note that if we calculate the elasticity for the same interval using a decrease in price from $3 to $2, we get (2 / 4) / ( $1 / $3) = 1.5, a different value. 0 $1 $2 $3 $4 $5 Price Insight For linear demand functions, elasticity changes at each point on the price-quantity demand function. LINEAR PRICE-QUANTITY DEMAND MBTN Management by the Numbers 14
Price Elasticity and Optimal Margin Example: Using the formula, q = 4 * p + 100, calculate the price elasticity at the point where p = $15, the profit-maximizing price. First, calculate q = 4 * 15 + 100 = 40 Next, the price elasticity is equal to the slope * p / q = 4 * (15 / 40) = 1.5 Now calculate the percentage margin on selling price at profitmaximizing price ($15 - $5) / $15 = 66.7% Divide the margin into 1 = 1 /.667 = 1.5 At the profit maximizing price, the elasticity is equal to the reciprocal of the margin and vice versa. The minus sign is ignored for these purposes. This is a very powerful and important result that always holds, regardless of the nature of the form of demand. PRICE ELASTICITY AND OPTIMAL MARGIN Definition At the Profit Maximizing Price: Elasticity = 1 / Margin%, and Margin% = 1 / Elasticity. MBTN Management by the Numbers 15
Selling Through Resellers When selling through resellers, we still need to calculate the MRP in terms of retail price, but now we have the added complication of the channel margins impacting the margin calculation. Since, retailers often require a percentage* margin, it is no longer a constant dollar variable cost as price changes. One way to handle this is to use the retail margin to convert Retail MRP to the Marketer MRP. Let s use the linear demand function below to illustrate. Example: Q = 4 * Retail Price + 100 and assume retailers earn a 40% margin and that unit costs are $5. First, calculate MRP = MWB/4 = 100/4 = $25 (Retail) Since retailers take 40% margins, our marketer only receives 60% of retail price. For her profit calculations, the MRP (Marketer) = $25 * 60% = $15. Maximum profit is earned at the Marketer price that is halfway between $15 and $5, or $10. This corresponds to a Retailer Price of $16.66 ($10/.6). SELLING THROUGH RESELLERS * If retailers use constant dollar margins, we could just add that dollar margin to our unit variable cost. MBTN Management by the Numbers 16
More on Price Elasticity Although price elasticity is not very useful if the demand function is linear, elasticity is the key to finding the optimal price for nonlinear demand curves. As we just observed, at the optimal price, the elasticity is always equal to the reciprocal of the margin (or vice versa) if the sign is dropped. We can easily verify this for our linear example. In Pricing II Constant Elasticity, we will examine demand curves that are not linear. For non-linear demand curves, it will not be as easy to find the optimal price. However, what we can do is compare the elasticity (if we know it at the current price) to the reciprocal of our current margin. If they are not equal, we know which direction to adjust our price to improve profit. Eventually, these adjustments will lead to the optimal price. MORE ON PRICE ELASTICITY MBTN Management by the Numbers 17
Further Reference Marketing Metrics by Farris, Bendle, Pfeifer and Reibstein, 2 nd edition, chapter 7. - And the following MBTN modules - Pricing II: Constant Elasticity - This module explores pricing under the assumption of demand curves that are not linear. Profit Dynamics - This module is a more basic module that examines price-volume trade-offs. Promotion Profitability - This module examines the role of promotion in pricing decisions. FURTHER REFERENCE MBTN Management by the Numbers 18