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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 Paul Farris, Phil Pfeifer and Management by the Numbers, Inc.

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2 Linear Price-Quantity Functions L INEAR P RICE -Q UANTITY F UNCTIONS This presentation will focus on linear demand functions where the relationship between price and quantity is linear. Definition 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.

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3 L INEAR P RICE -Q UANTITY D EMAND Linear Price-Quantity Demand 0$1$2$3$4$5 Price Quantity 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) / ( ). 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. Definition Slope of demand = change in quantity / change in price. Insight Price slope is almost always negative, but often people drop the sign.

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4 L INEAR P RICE -Q UANTITY D EMAND F ORMULA Linear Price-Quantity Demand Formula 0$1$2$3$4$5 Price Quantity 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. MWB -- Maximum Willing to Buy MRP -- Maximum Reservation Price MRP is where Q = 0. MWB is where P = 0

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5 L INEAR P RICE -Q UANTITY D EMAND F ORMULA Linear Price-Quantity Demand Formula Definitions MWB = Quantity - Slope * Price MRP = MWB / (- Slope) We took a bit of a leap of faith there. Lets make sure we dont lose you. You may recall from algebra (egad) 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. We can use this basic equation for price / quantity relationships as well as well as bringing in some other important managerial concepts such as the maximum willingness to buy (MWB) and maximum reservation price (MRP). First, lets express the basic linear function using marketing terminology: Quantity = Slope * Price + MWB or with a little substitution, Quantity = MWB * [1 – Price / MRP] From this, we can quickly derive functions solving for MWB and MRP as shown.

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6 O PTIMAL (P ROFIT M AXIMIZING ) P RICE Optimal (Profit Maximizing) Price 0$1$2$3$4$5 Price Quantity MWB MRP If you are told that the unit cost is $1, what is the optimal (profit-maximizing) price to charge? All of the information required is presented here Unit Cost

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7 O PTIMAL (P ROFIT M AXIMIZING ) P RICE Optimal (Profit Maximizing) Price 0$1$2$3$4$5 Price Quantity MWB MRP C= Unit Cost But, it is nice, and maybe a little surprising, that the profit-maximizing price is ALWAYS exactly half-way between these two points. 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

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8 B IG C ONCLUSION ! Big Conclusion! 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 just equal to 0) 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.

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9 E XAMPLE OF HOW TO C ALCULATE MWB AND MRP Example - How to Calculate MWB & MRP Say 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. This gives us two points to calculate the slope. Slope = delta Q / delta P = (8 - 12) / (5 - 4) = - 4 Using the Quantity = Slope * Price + MWB formula, substitute one set of price & quantity values from above along with the slope. 12 (quantity) = – 4 (slope) * 4 (price) + MWB, so MWB = = 28 Verify with the other point: 8 = 28 – 4 * 5 Now, it is easy to find MRP by solving for the price at which we sell zero units. 0 = 28 – 4 * price = $7. If our unit costs are $3, what would be the profit-maximizing price? Profit Maximizing Price = 1/2 (Cost + MRP) = ½ ( 3 + 7) = $5.

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10 Finding Optimal Price with Regression F INDING O PTIMAL P RICE WITH R EGRESSION Then add to the cost ½ the difference between cost and MRP Suppose we apply regression to sales data, and find the following demand function where slope = -4 and MWB = 100: If the unit cost equals $5, what is the optimal price? q = slope * p + MWB = – 4 * p First, find MRP by dividing MWB by – slope (e.g. a + value), 100 / 4 = 25, so MRP = 25 Cost + ½ (MRP – Cost) = $5 + ½ ($25 - $5) = $15 So, the profit-maximizing price is $15. Use Excel to test this.

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11 Slope versus Elasticity 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. S LOPE VERSUS E LASTICITY 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).

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12 Slope versus Elasticity S LOPE VERSUS E LASTICITY 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). Lets look at an example on the next slide.

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13 L INEAR P RICE -Q UANTITY D EMAND Linear Price-Quantity Demand 0 $1 $2 $3 $4 $5 Price Q 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. Insight For linear demand functions (fixed unit change in quantity for a given change in price), elasticity changes at each point on the price- quantity demand function.

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14 B ACK TO P RICE E LASTICITY Back to Price Elasticity 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 * = 40 Next, the price elasticity is equal to the slope * p / q = - 4 * (15 / 40) = Now calculate the percentage margin on selling price at profit-maximizing price…($15 - $5) / $15 = 66.7% Divide the margin into 1 = 1 /.667 = 1.5 At the optimal 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 results that always holds, regardless of the nature of the form of demand.

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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. Lets use the linear demand function below to illustrate. Q = -4*Retail Price 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). * If retailers use constant dollar margins, we could just add that dollar margin to our unit variable cost. S ELLING T HROUGH R ESELLERS

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16 M ORE ON P RICE E LASTICITY 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 part II, 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.

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