Basic Price Optimization OPRE 6377 Lecture Notes by Metin Çakanyıldırım Compiled at 15:03 on Tuesday 4 th September, 2018

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1 Basic Price Optimization OPRE 6377 Lecture Notes by Metin Çakanyıldırım Compiled at 5:03 on Tuesday 4 th September, 208 Willingness to Pay The frequency of customers willing to pay x in a population is denoted by w(x). Example: In a market study, 00 consumers are asked how much they are willing to pay for a local newspaper s Sunday edition. All of the respondents choose to pay between $0 and $2: Price to pay for the newspaper 0 2 # of consumers Frequency Hence, we set w(0) = 0., w() = 0.6 and w(2) = 0.3. Frequency w(x) can be interpreted as the probability that a randomly chosen consumer is willing to pay x. This implies that there is a willingness-to-pay (WTP) random variable. Example: The WTP random variable implied by the newspaper market study has the range of {0,, 2} and the probabilities of P(WTP = 0) = 0., P(WTP = ) = 0.6, P(WTP = 2) = 0.3. The willingness to pay (WTP) of a customer for a product of a company depends on. The price history of that product, 2. Prices currently charged for similar products by the company, 3. Prices currently charged for similar products by other companies. WTP of customers can be managed by explaining the value of a product to customers. Consider the case of drug coated coronary stent developed and sold by Johnson and Johnson (J&J) in This stent was priced at $3500 per piece while the regular stents were priced about $000 per piece. In other words, the new stent was priced 250% more than the regular stent. J&J customers (doctors and patients) criticized this price and called it price gouging. Clearly, they were not willing to pay $3500 for the new stent. To pull WTP up, J&J started to communicate the economic benefits of the new stent to the customers. Two numbers are used in the economic benefit analysis: drug coated stent reduces the chances of a repeated operation from 20% to 5% in a year and the cost of an operation runs about $30,000. Putting these numbers together, the new stent offers a value of 0.5(30,000)= $4500. With this economic analysis, J&J answered the critics and pulled the WTP up. Since Pharmaceutical companies frequently bring new drugs into the marketplace and they would like to set high prices to compensate for R&D costs, these companies can usually find it difficult to reconcile the price of a product with the WTP of the customers. They can also use the tactic of communicating the economic value to customers, as it is exemplified by J&J. An important concept is the reference price. This is the price that we associate with an aggregate group of products. It is rather a rough estimate, one that is made without knowing the detailed properties of the

2 product. This estimate is certainly shaped by the factors that affect the WTP: price history, price of similar products, price of competitor s products, value of the product. Example: Consider how much textbooks cost nowadays. Certainly they cost much more than what they should. Even from this last statement, we can infer that we have an expectation for the price of a textbook. For example, your reference price can be $50 which could be based on the prices of the latest textbooks that you bought. Then we say that your reference price for a textbook is $50. If you are a student of operations management, you often buy operations books that may cost around $20. On the other hand, if you are pursuing an accounting degree, you often buy accounting books that may cost around $250. In a particular OPRE 6377 section, there are 20 operations and 6 accounting students, what would be the average WTP for this section? To answer this we take a weighted average: = = ( ) = = A customer s WTP is strongly related to the reference price. To increase WTP, we can increase the reference price. This can be done by pointing out and emphasizing expensive products. For example, your reference price for a textbook can be increased by telling you that a graduate level accounting textbook costs $250. This consciously or unconsciously will force you to revise your reference price, or at least it will make you uncertain of your reference price. After announcing the accounting textbook s price, we can look at the price of the textbook for OPRE The list price is $64 and the current discounted Amazon price (at the time of writing this example) is $50. Since the accounting textbook costs 380% more than the OPRE 6377 textbook and the high price of accounting textbooks have already pulled your WTP up, you are willing to pay $50 to the OPRE 6377 textbook. A bookstore clerk who wants to sell OPRE 6377 textbook should then first tell the customers the prices of the accounting textbooks. This tactic of starting with a higher priced item to pull WTP (or reference price) up is called top-down selling. Another tactic is giving 00% of something to a customer but taking a smaller percentage of another thing from the customer. You give 00% of something small and take a smaller percentage of something much bigger. As an example consider free Internet access at premium hotels. The cost of providing free Internet access is much less than the premium the hotels charge for it. This extra charge may be increasing the hotel bill by $0, which is 5% of $200 average nightly rate at a premium hotel. In this case hotel bills 5% more to the customer but gives 00% free Internet service. Many customers prefer to pay 5% more in their bill in return for 00% free Internet service. This happens because of our habit of focusing on percentages rather than absolute values when we compare prices. Consider the stent and textbook examples above, if you have paid more attention to percentages of 250% and 380% than the absolute differences of $2500 and $200, you also have the habit of focusing on the percentages. In order to highlight our focus on percentages, consider a $0 saving that you can make by switching where you are shopping for clothes. When you are spending less than $40 (more than 25% saving) you are more likely to switch the location than when you spend more than $00 (less than 0% saving). This has been established with customer surveys and experiments. Roughly speaking, we are less concerned about small savings when what we are buying is big or important. For example maintenance costs of a car is not a deal maker or breaker when buying a car. Another example is the cost of the restaurant that you will have your birthday party, the saving that you can have by going to a cheap restaurant does not sound interesting, does it? The percentage of consumers buying a product at price p are the ones who are willing to pay price p or more: P(A random consumer purchases at price p) = P(WTP p) = P(WTP p 0). WTP p is referred to as surplus obtained by purchasing the product. Clearly, a consumer does not purchase a product that provides a negative surplus. 2

3 Since the purchase probability requires assessing the frequency of consumers that fall into the range of [p, ), we can represent it by writing for discrete valued WTPs and P(A random consumer purchases at price p) = P(A random consumer purchases at price p) = p w(x)dx w(i) i=p for continuous valued WTPs. If we consider the probability of no-purchase, we need to assess the frequency of consumers that fall into the range of [0, p). But these probabilities are given in terms of the cumulative distribution function (cdf) W of the WTP random variable. For continuous valued WTP, P(A random consumer does not purchase at price p) = For discrete valued WTP, P(A random consumer does not purchase at price p) = p 0 w(x)dx = P(WTP p) =: W(p). p w(i) = P(WTP p ) =: W(p ), i=0 where p denotes the largest price smaller than p. When prices take continuous values, p = p. When prices take integer values, p = p. But prices are not always integer valued or do not always have a unimodal WTP distribution. Example: The WTP for a bottle of water is given as Price p to pay for a bottle of water in $ Probability w(p) C.d.f. W(p) What are (0.93) and (.05)? What are the probabilities that a random customer does not purchase at prices $0.93 and $.05? The highest price smaller than 0.93 is 0.90, so (0.93) = The highest price smaller than.05 is.00, so (.05) =.00. Note that P(A random consumer does not purchase at price 0.93) = W(0.90) = P(A random consumer does not purchase at price.05) = W(.00) = Also note that w(p) has two modes at $0.97 where w(0.97) = 0.32 and at $.0 where w(.0) = 0.4. The WTP distribution does not have to be unimodal. The above arguments yield that the portion of the consumers willing to pay price p or more is P(Purchase at price p) = W(p) = W( ) W(p) = Following two equalities used to relate W and w W(p) = p 0 0 w(x)dx w(x)dx and w(x) = d dx W(x). 3 p 0 w(x)dx = p w(x)dx

4 W(p) is the portion of the customers who are willing to pay at most p, that is they are willing to pay p or lower. This portion does not buy the product if it is priced slightly above p. These are relations for continuous prices, their analogs for discrete prices are P(Purchase at price p) = W(p ) = W( ) W(p ) = Following two equalities used to relate W and w W(x) = x i=0 i=0 w(i) w(i) and w(x) = W(x) W(x ). p i=0 w(i) = w(i) i=p We call W as the cumulative willingness to pay and w as the exact willingness to pay. Often such a distinction is not necessary as W and w are closely associated with each other through the formulas above. 2 Market Size The number of consumers who are considering to buy a product in a market is termed as market size and is denoted by D. We can assume that each of these consumers independently decide to purchase or not, then the number of consumers who decide to purchase has a binomial distribution with parameters D and W(p). You can think of each consumer as an experiment, W(p) as the success probability of each experiment, then the number of successes will have binomial distribution with parameters (D, W(p)). The expected number of purchases is the expected value of the binomial distribution, which is Expected number of purchases = D( W(p)). Although we have interpreted the number of purchases and willingness to pay as random variables to arrive at D( W(p)) expression, the demand is often directly written as d(p) = D( W(p)). The probabilistic interpretation provides more insight and richness, whereby any cdf can be a candidate for W. 3 Logit Price Response Function An often used price response function is the logit price response, which has two parameters a, b, and W(p) = + exp(a bp) for p. Note that W( ) = 0 and W( ) =. A higher price p must reduce the likelihood of a purchase, that is W(p) must decrease in p. Technically, the derivative of W(p) must be nonnegative: w(p) = d dp W(p) = b exp(a bp) ( + exp(a bp)) 2 0, which implies b 0. Many consumers have willingness to pay around a reference price. The w(p) function is maximized around the reference price. To find the maximizer of w(p), we consider the derivative of w(p): d dp w(p) = b2 exp(a bp)( + exp(a bp)) 2 + b 2 exp(2(a bp))2( + exp(a bp)) ( + exp(a bp)) 4. 4

5 Setting the derivative equal to zero gives 0 = b 2 exp(a bp)( + exp(a bp)) 2 + b 2 exp(2(a bp))2( + exp(a bp)) ( ) = b 2 exp(a bp)( + exp(a bp)) ( + exp(a bp)) + 2 exp(a bp). This equation has a single root that satisfies = exp(a bp) or p = a/b. Hence, the reference price implied by the logit response is a/b. Often this price is nonnegative and a 0. Note that a 0 is possible but rare. On the other hand, b is always nonnegative from above. Interestingly, w(p) is symmetric around its maximizer (the reference price) a/b: w(p = a/b x) = w(p = a/b + x) b exp(a b(a/b x)) ( + exp(a b(a/b x)) 2 = b exp( bx) ( + exp( bx)) 2 = b exp(a b(a/b + x)) ( + exp(a b(a/b + x)) 2 b exp(bx) ( + exp(bx)) 2, which follows from ( + exp(bx)) 2 = exp(2bx)( + exp( bx)) 2. The growth w(p) in W(p) is the same at the price p and p 2 provided that (p /p 2 )/2 = a/b. Example: The willingness-to-pay (WTP) for a lunch meal can have a logit function with a = 5 and b = 0.5, W(p) = + exp(5 p/2) and w(p) = exp(5 p/2) 2 ( + exp(5 p/2)) 2 Then we know that w(p) is maximized at $0 = a/b, so the highest percentage of customers is willing to pay $0 in the table below. p w(p) W(p) W(p) p w(p) W(p) W(p) p w(p) W(p) W(p) Mode of $0 does not say that most customers are willing to pay $0. Despite w(0) = = /8 > w(p) for p {4, 5,..., 6} in the table, /8 of the customers cannot be called the most. The probability density function (pdf) corresponding to the logit price response is w(p) = b exp(a bp) ( + exp(a bp)) 2 = e p a/b /b ) (/b) ( + e p a/b 2. /b This pdf is known as logistics distribution with mean a/b and scale parameter /b. The variance is π 2 /(2b 2 ). Fitting logistics distribution to data {(p i, w(p i ))} to estimate a, b is known as logistics regression. 5

6 4 Multinomial Choice Model In the multinomial model, demand of a product is proportional to that product s attractiveness while it decreases with other products attractiveness. This is a reasonable method to split demands among products. But it requires the computation of attractiveness of a product. The multinomial model can be thought as a hierarchical method of assigning demands to products. This hierarchy has two steps: computation of attractiveness of each product and splitting demand among products depending on relative attractiveness. Attractiveness: We consider only price as a variable although one can append quality, functionality or other aspects of a product to this framework. Suppose that products are indexed by letter i, the price of product i is p i and its attractiveness is A i (p i ). Attractiveness is nonnegative, i.e., A i (p i ) 0. Since higher prices make a product less attractive, A i must be a decreasing function. This is not a very workable construction until we make A i (p i ) more explicit. In the multinomial model, we set A i (p i ) = exp( b i p i ) by using parameter b i 0. This parameter is a measure of drop in attractiveness as the price increases. The drop in attractiveness is proportional to the parameter b i and the current attractiveness. To make this more concrete suppose that p i is the current price and it yields the attractiveness A i (p i ) and further suppose that the price is increased to p i > p i. The attractiveness at the higher price p i is A i(p i ). Since A i is a decreasing function we have A i (p i ) < A i(p i ). The quantity of interest is incremental change in the attractiveness A i (p i ) A i(p i ). In the multinomial model, this incremental change is given by A i (p i ) A i(p i ) b i A i (p i ) < 0. In words, the attractiveness drops faster with price increases if the parameter b i is large or the current attractiveness is large. If we write the above approximate equality in terms of a derivative, we obtain d dp i A i (p i ) = b i A i (p i ). The only attractiveness function that solves this equality is A i (p i ) = exp( b i p i ), which is the multinomial model. Relative Attractiveness: Suppose that the market has I products, each product i has its price p i and attractiveness A i (p i ) as formulated above. The relative attractiveness f i for product i depends on the price vector, i.e., all of the prices p = [p, p 2,..., p i,..., p I ]. Algebraically, it is given by f i (p) = f i ([p, p 2,..., p i,..., p I ]) = A i(p i ) I i= A i(p i ). Since f i is relative attractiveness, it is the ratio of product i s attractiveness to the sum of all products attractiveness. Figuratively speaking, you can think of product i trying to overcome the attractiveness of all of the other products to be sold to the customer. This ratio automatically gives us the following four properties: f i (p) 0; d dp i f i (p) 0; d dp j f i (p) 0; I f i (p) =. i= Attractiveness and relative attractiveness concepts are more general than the multinomial model. Put differently, the multinomial model is a special case of attractiveness and relative attractiveness concepts. These concepts together are called attraction models. Example of alternative attractiveness: The multinomial model has exponential attractiveness A i (p i ) = exp( b i p i ). By changing the attractiveness function, we can obtain another model. We can consider 6

7 A i (p i) = b i /p i as an alternative. This is a valid alternative because A i is positive and decreasing in price. Then, for three products, we have the relative attractiveness of f (p) = b /p b /p + b 2 /p 2 + b 3 /p 3 ; f 2 (p) = b 2 /p 2 b /p + b 2 /p 2 + b 3 /p 3 ; f 3 (p) = b 3 /p 3 b /p + b 2 /p 2 + b 3 /p 3. Example of multinomial with power-functions for attractiveness: A valid question is what happens if we set a slightly different attractiveness A i (p i) = 3 b i p i rather than A i (p i ) = exp( b i p i ) = e b i p i. 3 b i p i and e b i p i are quite alike but can they give us the same attractiveness? The answer is yes because A i (p i) = 3 b i p i = e b i p i = A i (p i ) provided that b i ln 3 = b i, where ln denotes natural logarithm. Consequently, we obtain the same multinomial demand models whether we set A i (p i) = 3 b i p i or A i (p i ) = e b i p i when b i ln 3 = b i. Parameters are estimated in practice, if you work with A i (p i) = 3 b i p i, you will estimate ˆb i from the historical data. Alternatively, if you work with A i (p i ) = e b i p i, you will estimate ˆb i from the historical data. Except for small estimation errors, you will obtain ˆb i ln 3 = ˆb i. In other words, you gain or lose nothing by considering the alternative attractiveness formulation A i (p i) = 3 b i p i over the original formulation A i (p i ) = e b i p i. Above argument can be generalized for any alternative attractiveness formulation with power functions: A i (p i) = c b i p i, where c is a given constant. Example: Consider the multinomial choice model with two products 0 and where b 0 = 0 and b = b > 0. Find the attractiveness and relative attractiveness of the products at p = [p 0, p ]. A 0 (p) = e 0p 0 = and A (p) = e bp, moreover f 0 ([p 0, p ]) = A 0 (p) A 0 (p) + A (p) = + e bp and f ([p 0, p ]) = A (p) A 0 (p) + A (p) = e bp + e bp. If we interpret product 0 as no purchase and set p 0 = 0, the relative attractivenesses above are the nopurchase and purchase probabilities of product with logit response function and with parameters a = 0 and b. 5 Homework Questions. [Top-Down Selling] Top-down selling tactic is initially offering high-priced items to a customer to pull WTP (reference price) up regardless of whether the customer is interested in or is able to afford these high-priced items. Suppose that you are a) a real estate agent selling houses, b) a mail catalog designer for a blouse company, c) a Wal-Mart store employee stocking pasta shelves. How would you implement the top-down selling tactic in two of the three contexts given above? 2. [Less rent or free month(s)] When the apartment rental market cools down, the apartment owners think of reducing the rental prices either directly or indirectly to stimulate the demand. Direct reduction is simply reducing the annual rent. One of the indirect reduction strategies is keeping the rent constant but not charging any the rent for the first month of a 2-month lease (also called free first-month). Among these two strategies, which one keeps the reference price higher? Explain. Is it in the interest of the apartment owners to keep the reference price higher? In your own apartment renting experience, have you seen more direct rent reductions or free first-months? 7

8 3. [Inattention bias] It is empirically observed that the price of a car decreases with its mileage. However, this decrease is not continuous at every mileage. For example, the decrease in price with extra mile is significant if the car already has 9,999 miles. This decrease is small if the car has 9,998 miles. In general, price drops by a significant amount when it reaches mileage of 20 K, 30 K, 40 K, 50 K, and so on. These drops are attributed to inattention bias of the buyers. In other words, buyers typically pay attention only to the first digit of the mileage for the cars below 00 K. Thus, cars with 6000 miles and with miles are perceived to closer in terms of mileage than cars with miles and with 7000 miles. The inattention bias can be strengthen in a sales transaction by increasing the amount of information the buyer receives. a) If you are a car salesperson and want to sell a car with 5023 miles on it, would you plan to strengthen the inattention bias or weaken it? b) How would your response to a) change if the car has miles? c) What are the specific ways to strengthen the inattention bias that a car salesperson can implement? 4. [Discrete-valued WTP of a Coffee Mug] The discrete-valued WTP for a coffee mug is given as Price p to pay for a coffee mug in $ Probability w(p) C.d.f. W(p)????????? a) Complete the table by finding the cumulative WTP distribution W(p) at each price p. b) What is (9), the largest WTP smaller than $9? What is (9.70)? c) What are the probabilities that a random customer purchases at prices $9 and $9.7? 5. [Revenue from a Coffee Mug with Discrete-valued WTP] WTP for a coffee mug is Price p to pay for a coffee mug in $ Probability w(p) C.d.f. W(p) W((p) )????????? Revenue p( W((p) ))????????? a) Complete the table by finding the purchase probability W((p) ) at each price p. b) Complete the table by finding the expected revenue pp(wtp p) = p( W((p) )) from a single customer at each price p and report the revenue-maximizing price. 6. [Continuous-valued WTP] Suppose that a product has a reference price of ˆp and willingness to pay is uniform over the interval [ ˆp, ˆp + ] for a known > 0: 0 if x < ˆp, w(x; ˆp) = if ˆp x ˆp +, 2 0 if ˆp + < x a) For each of the three possible cases of price p, p < ˆp, ˆp p ˆp + and ˆp + < p, express the percentage of the customers who are willing to pay at most p in terms of ˆp, and numbers. b) For each of the three possible cases of price p, p < ˆp, ˆp p ˆp + and ˆp + < p, express the percentage of the customers who are willing to pay at least p in terms of ˆp, and numbers.. 8

9 c) Suppose that a top-down selling tactic increases the reference price from ˆp to ˆp + but it does not change the uniformity of WTP or the width of the WTP interval, express the new WTP w(x; ˆp + ) in terms of ˆp, and numbers. How much does this tactic increase the percentage of the population who buy the product at price p for ˆp + p ˆp +? Express the increase in terms of ˆp, and numbers. 7. [M-shaped WTP] The willingness-to-pay density for a shirt is positive only over [80, 20] and is 20 p 80 if 80 p 00, 400 w(p) =. p 00 if 00 p 20, 400 a) Draw the WTP density and show that it is made up of two triangles touching only at a single point, the price of $00. What are the heights of these two traingles? b) Find the cumulative WTP distribution W(p). 8. [Revenue from a product with Continuous-valued WTP] The demand for a product is given by 000 p d(p) = p (0 p)2 2 if p 0, if 0 p 20, if 20 p a) Find d(p = 9.5) and d(p = 0.5). Do the numerical values obtained for d(p = 0) via 000/p and 00 (0 p) 2 /2 concide? If yes, we say d(p) is continuous at p = 0. b) Consider the revenue function R(p) = pd(p). What are the revenue maximizing prices over the ranges of p [0, 0] and p [20, )? c) The revenue function over the range of p [0, 20] is one of the following:. R(p) = p 3 + 5p p R(p) = p 3 /2 + 5p p R(p) = p 3 /2 + 0p p R(p) = p 3 + 0p p Identify the corect revenue function and take its derivative to find the quadratic function: What are the values of a, b, c? R (p) = ap 2 + bp + c. d) Solve R (p ) = 0 for p [0, 20] to find p and comment whether the revenue maximizing price is p or it is less than 0 or more than [Elasticity] From the class discussion we know that the elasticity ϵ(p) of price response satisfies ϵ(p) = p ( ) d d(p) dp d(p). a) Obtain a simpler elasticity expression for the demand model d(p) = D( W(p)) where D is the market size and W is the cdf for continuous valued WTP whose pdf is w. Observe if the elasticity depends or does not on the market size. 9

10 b) Suppose that the WTP has the logistics distribution with parameters a, b, i.e., price response has logit function W(p) = for p. + exp(a bp) Specialize the elasticity expression of a) for the logit response function. c) Compute the elasticity in b) for a = 3, b = 0.3 and p = 0 0. [Multinomial Logit] There are two options to go from UTD campus to the DFW airport: Taxi and Uber. The prices respectively are p T = $60 and p U = $40 for a taxi and Uber ride. We also have parameters b T = 0.0 and b U = a) Find attractivenesses of both taxi and Uber options: exp( b T p T ) and exp( b U p U ). Given that one has a flight departing from the DFW airport, he must go to the airport and so staying at UTD is not a choice. Find the choice probabilties for taxi and Uber rides. b) What happens to choice probabilities when taxis and Uber discounts their prices respectively by $20 and $0? c) The amount of price changes from a) to b) are in the form of p T + δ and p U + (b T /b U )δ where δ = 20. For a general δ > 0 write the choice probability for Uber and relate this to the same probability under δ = 0. Does the choice probability depend on δ? 0