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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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**Linear Price-Quantity Functions**

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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**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) / ( ). 10 9 8 7 6 5 4 3 2 1 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. Quantity Insight Price slope is almost always negative, but often people drop the sign. 0 $1 $2 $3 $4 $5 Price Definition Slope of demand = change in quantity / change in price.

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**Linear Price-Quantity Demand Formula**

MWB is where P = 0 MWB -- Maximum Willing to Buy 10 9 8 7 6 5 4 3 2 1 Formulas for linear demand functions use the following format (recall y = mx + b): Quantity = Slope * Price + MWB In this example, beta = One could then solve for MWB and MRP using this equation and a price quantity point. Quantity MRP -- Maximum Reservation Price 0 $1 $2 $3 $4 $5 Price MRP is where Q = 0.

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**Linear Price-Quantity Demand Formula**

We took a bit of a leap of faith there. Let’s make sure we don’t 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, let’s express the basic linear function using marketing terminology: Quantity = Slope * Price + MWB or with a little substitution, Quantity = MWB * [1 – Price / MRP] Definitions MWB = Quantity - Slope * Price MRP = MWB / (- Slope) From this, we can quickly derive functions solving for MWB and MRP as shown.

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**Optimal (Profit Maximizing) Price**

MWB 10 9 8 7 6 5 4 3 2 1 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 Quantity MRP 0 $1 $2 $3 $4 $5 Price Unit Cost

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**Optimal (Profit Maximizing) Price**

As you might expect, the profit maximizing price is always more than cost and less than the MRP. MWB 10 9 8 7 6 5 4 3 2 1 But, it is nice, and maybe a little surprising, that the profit-maximizing price is ALWAYS exactly half-way between these two points. Quantity Profit = Q * (P-C) = 4 * ($3-$1) = $8 MRP 0 $1 $2 $3 $4 $5 Price C= Unit Cost

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Big Conclusion! 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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**Example of how to Calculate MWB and MRP**

Example - How to Calculate MWB & MRP Example of how to Calculate MWB and 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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**Finding Optimal Price with Regression**

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? 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. Use Excel to test this.

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**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.

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**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). Let’s look at an example on the next slide.

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**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 10 9 8 7 6 5 4 3 2 1 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 Q 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. $ $ $ $ $5 Price 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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**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) = - 1.5 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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**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. 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.

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**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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