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I am in need of help with a discussion forum that involves supply chain within amazon. I hace attached the question and the reading that covers it. If you have any questions or concerns please let me know.
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Please answer the following questions after reviewing the reading and lessons for Week 7.
What measures would you use in a business like Amazon.com to evaluate the company’s
performance? The supply chain?
Response must be minimum 250 words
3
basic price optimization
In this section we introduce the basic elements of pricing and revenue optimization and
show how the basic pricing revenue and optimization problem can be formulated as an optimization problem. The goal of the optimization problem—its objective function—is to
maximize contribution: total revenue minus total incremental cost from sales. The key elements of this problem are the price-response function and the incremental cost of sales,
both of which we introduce in this section. We then formulate and solve the pricing and revenue optimization function in the case of a single product in a single market without supply constraints and derive some important optimality conditions.
Copyright © 2005. Stanford University Press. All rights reserved.
3.1 the price-response function
A fundamental input to any PRO analysis is the price-response function, or price-response
curve, d(p) which specifies how demand for a product varies as a function of its price, p.
There is one price-response function associated with each element in the PRO cube—that
is, there is a price-response function associated with each combination of product, marketsegment, and channel. The price-response function is similar to the market demand function found in economic texts. However, there is a critical difference. The price-response
function specifies demand for the product of a single seller as a function of the price offered by
that seller. This contrasts with the concept of a market demand curve, which specifies how
an entire market will respond to changing prices. The distinction is critical because different firms competing in the same market face different price-response functions. Referring
to Table 1.2, the price-response function facing Amazon for Bag of Bones is likely to be quite
different from the one facing ecampus.com. The differences in the price-response functions
faced by different sellers are the result of many factors, such as the effectiveness of their marketing campaigns, perceived customer differences in quality, product differences, and location, among other factors.
In a perfectly competitive market, the price response faced by an individual seller is a vertical line at the market price, as shown in Figure 3.1. If the seller prices above the market
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
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Demand
basic price optimization
39
D
d(p)
0
P
Price
Figure 3.1 Price-response curve in a perfectly competitive market.
price, his demand drops to 0. If he prices below the market price, his demand is equal to the
entire market. A standard example used in economics texts is wheat:
Copyright © 2005. Stanford University Press. All rights reserved.
The best example to keep in mind is that of a wheat farmer, who provides a minuscule
percentage of the wheat grown in the world. Regardless of whether he produces 10 bushels or
1,000, he remains too small to have any impact on the going market price. . . . If he tries to
charge even a fraction of a penny more, he will sell no wheat, because buyers can just as easily
buy from someone else. If he charges even a fraction of a penny less, the public will demand
more wheat from him than he can possibly produce— effectively an infinite quantity.1
In other words, wheat is a commodity—buyers are totally indifferent among the offerings
of different sellers, they have perfect knowledge about all prices being offered, and they will
buy the product only from the lowest-price seller. Furthermore, each seller is small relative
to the total size of the market. In this situation, the seller has no pricing decision—his price
is set by the operation of the larger market. To quote a popular text: “In a competitive market, each firm only has to worry about how much output it wants to produce. Whatever it
produces can only be sold at one price: the going market price.” 2 At any price below the
market price, the demand seen by a seller would be equal to the entire demand in the market—the amount D in Figure 3.1. At any price above the market price he sells nothing. The
seller of a true commodity in a perfectly competitive market has no need for pricing and
revenue optimization—indeed, he has no need of any pricing capability whatsoever. However, true commodities are surprisingly rare. The vast majority of companies face finite customer responses to price changes and therefore have active PRO decisions.
The price-response functions that we consider—and those facing most companies most
of the time— demonstrate some degree of smooth price response. An example is shown in
Figure 3.2. Here, as price increases, demand declines until it reaches zero at some satiating
price P. This type of smooth market-response function is usually termed a monopolistic or
monopoly demand curve in the economics literature. The terminology is somewhat unfortunate, since the fact that a company faces some level of price response hardly means that
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
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basic price optimization
Demand
D d(0)
d(p)
0
P
Price
Figure 3.2 Typical price-response curve.
it is a “monopoly.” Companies such as United Airlines, UPS, and Ford all face smooth
price-response functions for their products, yet none of them would be considered a monopoly in the usual sense of the world.
Copyright © 2005. Stanford University Press. All rights reserved.
3.1.1 Properties of the Price-Response Function
The price-response functions used in PRO analysis have a time dimension associated with
them. This is in keeping with the dynamic nature of PRO decisions—we are not setting
prices that will last “in perpetuity” but prices that will be in place for some finite period of
time. The period might be minutes or hours (as in the case of a fast-moving e-commerce
market), days or weeks (as in retail markets), or longer (as in long-term contract pricing).
At the end of the period we have the opportunity to change prices. The demand we expect
to see at a given price will depend on the length of time the price will be in place. Thus, we
can speak of the price-response function for a model copier over a week or over a month,
but without an associated time interval there is no single price-response function.
There are many different ways in which product demand might change in response to
changing prices and, thus, many different possible price-response functions. However, all
of the price-response functions we consider will be nonnegative, continuous, differentiable,
and downward sloping.
Nonnegative. We assume that all prices are greater than or equal to zero; that is, p 0.3
Continuous. We assume that the price-response function is continuous—there are no
“gaps” or “jumps” in the market response to our prices. More formally, if d(0) D and P
is the satiating price, that is, the lowest price for which d(P) 0, then for every 0 q D
there is a price p P such that d(p) q. This implies that there is a price that will generate
any level of demand between 0 and the maximum demand. This property, known as
invertability, is often very useful.4
Differentiable. Differentiability means that the price-response function is smooth, with
a well-defined slope at every point.5 As with the assumption of continuity, this assumption
involves taking a mathematical liberty, since prices are only defined for fixed increments.
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
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basic price optimization
41
However, differentiability allows us to use the tools of calculus to solve the constrained optimization problems that arise in PRO—a gain that outweighs the slight imprecision that
results from using derivatives rather than difference equations.
Downward sloping. We assume that d(p) is downward sloping whenever d(p) 0 —
that is, that raising prices for a good during a period will decrease demand for that good during that period unless demand is already 0, in which case demand will remain at 0. Conversely, lowering prices can only increase demand.
The “downward sloping” assumption calls for a bit more discussion. First of all, it should
be noted that downward-sloping demand curves do not mean that high prices will always
be associated with low demand. A hotel revenue management will experience higher average rates when occupancy is high and lower average rates when occupancy is low. What the
downward-sloping property does indicate is that, in any time period, demand would have
been lower if prices had been higher, and vice versa. This corresponds both to economic
theory (in which consumers maximize their utilities subject to a budget constraint) and to
real-life experience.
Nonetheless, there are at least three cases in which the downward-sloping property may
not hold.
Copyright © 2005. Stanford University Press. All rights reserved.
1. Giffen goods. Economic theory allows for the possibility of so-called Giffen goods,
whose demand rises as their price rises because of substitution effects. An example
might be a student on a strict budget of $8.00 per week for dinner. When hamburger
costs $1.00 per serving and steak costs $2.00, he eats hamburger six times a week
and steak once. If the price of hamburger rises to $1.10, to stay within his budget,
he stops buying steak and buys hamburger seven times a week. In this case, a rise in
the price of hamburger causes his consumption of hamburger to increase. While this
behavior might conceivably occur at an individual level, a Giffen good requires that
enough buyers act this way that they overwhelm other buyers who would buy less
hamburger as the price rises. Giffen goods are almost never encountered in reality—
in fact, many economists doubt whether they have ever existed.
2. Price as an indicator of quality. In some markets, price is used by some consumers
as an indicator of quality: Higher prices signal higher quality. In this case, lowering
the price for a product may lead consumers to believe that it is of lower quality,
and demand could drop as a result. Typically, markets where this is an issue have a
large number of alternatives and some “lazy” buyers who do not have the time or
resources to research the relative quality of all the alternatives so that they use price
as a proxy. Wine is a classic example: Faced with a daunting array of labels and varietals, many purchasers are likely to use a rule such as: a $10 bottle for dinner with
the family, a $15 bottle if the couple next door is dropping by, and a $25 bottle if our
wine-snob friends are joining us for dinner.6 The “price-as-an-indicator-of-quality
effect” can be particularly important when a new product enters the market. A
medical-product company developed a way of producing a home testing device
at a cost 75% below the cost of the prevailing technology. They introduced the new
product at a list price 60% lower than the list price of the leading competitors, expecting to dominate the market. When sales were slow, they repackaged the product
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
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basic price optimization
and sold it at a price only 20% lower than the leading competitor. This time sales
took off. Their belief is that the initial rock-bottom price induced customers to believe their product was inferior and unreliable. The higher price was high enough
not to raise quality concerns but low enough to drive high sales.
3. Conspicuous consumption. Thorstein Veblen coined the term conspicuous consumption for the situation in which a consumer makes a purchase decision in order to
advertise his ability to spend large amounts. It probably does not come as a shock that
the reason some rock stars drink $300 bottles of Cristal champagne and drive Bentleys is not their finely honed appreciation of fine French champagne and British automotive engineering. Conspicuous consumption postulates a segment of customers
who buy a product simply because it has a high price—and others know it. Dropping
the price in this case may cause the product to lose its cachet and decrease demand.
While duly noting these three exceptions to downward-sloping demand curves, we will
proceed to ignore them for the remainder of the book. In defense of this decision, we observe that for almost all items, almost all of the time, raising the price will lower demand and
lowering the price will increase demand.
3.1.2 Measures of Price Sensitivity
It is often useful to have a simple characterization of the price sensitivity implied by a priceresponse function at a particular price. The two most common measures of price sensitivity are the slope and the elasticity of the price-response function.
Slope. The slope of the price-response function measures how demand changes in response to a price change. It is equal to the change in demand divided by the difference in
prices, or
Copyright © 2005. Stanford University Press. All rights reserved.
d 1p2, p1 2 3d 1p2 2 d 1p1 2 4/1p2 p1 2
(3.1)
By the downward-sloping property, p 1 p 2 implies that d(p 1) d(p 2). This means that
d(p 1, p 2) will always be less than or equal to zero.
The definition in Equation 3.1 requires two prices to be specified, because the slope of a
price-response function will be constant across all prices only if it is linear. However, it is
common to specify the slope at a single price, say, p 1, in which case it can be computed as
the limit of Equation 3.1 as p 2 approaches p 1. That is,
d 1p1 2 lim 3d 1p1
hS0
d¿ 1p1 2
h2 d 1p1 2 4/h
where d(p 1) denotes the derivative of the price-response function at p 1. By the differentiability property, we know that this derivative exists. The downward-sloping property means
that the slope will be less than or equal to zero for all prices.
The slope can be used as a local estimator of the change in demand that would result from
a small change in price. For small changes in price, we can write
d 1p2 2 d 1p1 2 d 1p1 2 1p2 p1 2
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
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(3.2)
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43
That is, a large (highly negative) slope means that demand is more responsive to price than
a smaller (less negative) slope.
Example 3.1
The slope of the price-response function facing a semiconductor manufacturer
at the current price of $0.13 per chip is 1,000 chips/week per cent. From
Equation 3.2, he would estimate that a 2-cent increase in price would result in
a reduction in demand of about 2,000 chips per week and a 3-cent decrease in
price would result in approximately 3,000 chips/week in additional demand.
It is important to recognize that the quality of the approximation in Equation 3.2 declines for larger changes in prices and that the slope cannot be used as an accurate predictor of demand at prices far from the current price. It is also important to realize that the
slope of the price-response function depends on the units of measurement being used for
both price and demand.
Example 3.2
The price of a bulk chemical can be quoted in either cents per pound or dollars
per ton. Assume that the demand for the chemical is 50,000 pounds at 10 cents
per pound but drops to 40,000 pounds at 11 cents per pound. The slope of the
price response function at these two points is
d 110, 112 150,000 40,0002 / 110 11 2
10,000 pounds/cent
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The same slope in tons per dollar would be (25 20)/(0.1– 0.11) 500 tons/
dollar.
Price elasticity. Perhaps the most common measure of the sensitivity of demand to price
is price elasticity, defined as the ratio of the percentage change in demand to the percentage
change in price.7 Formally, we can write
P 1p1, p2 2
1005 3d 1p2 2 d 1p1 2 4 /d 1p1 2 6
10051p2 p1 2 /p1 6
(3.3)
where P(p 1, p 2) is the elasticity of a price change from p 1 to p 2. The numerator in Equation
3.3 is the percentage change in demand, and the denominator is the percentage change in
price. Reducing terms gives
P 1p1, p2 2
3d 1p2 2 d 1p1 2 4 p1
3p2 p1 4d 1p1 2
(3.4)
The downward-sloping property guarantees that demand always changes in the opposite
direction from price. Thus, the minus sign on the right-hand side of Equation 3.4 guarantees that P(p 1, p 2) 0. An elasticity of 1.2 means that a 10% increase in price would result in
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
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basic price optimization
a 12% decrease in demand and an elasticity of 0.8 means that a 10% decrease in price would
result in an 8% increase in demand.
P(p 1, p 2) as defined in Equation 3.4 is sometimes called the arc elasticity. That it requires
two prices to calculate reflects that the percentage change in demand resulting from changing prices will depend on both the old price and the new price. In fact, the percentage decrease in from a 1% increase in price will generally not even be the same as the percentage
increase in demand that we would experience from a 1% decrease in price. For this reason,
both prices need to be specified in order to fully characterize elasticity. However, as we did
with slope, we can derive a point elasticity at p by taking the limit of Equation 3.4 as p 2
approaches p 1:
P 1p1 2 d¿1p1 2p1/d 1p1 2
(3.5)
In words, the point elasticity is equal to 1 times the slope of the demand curve times the
price, divided by demand. Since d(p) 0, the point elasticity P(p) calculated by Equation
3.5 will be greater than or equal to zero. The point elasticity is useful as a local estimate of
the change in demand resulting from a small change in price.
Example 3.3
A semiconductor manufacturer is selling 10,000 chips per month at $0.13 per
chip. He believes that the price elasticity for his chips is 1.5. Thus, a 15% increase in price from $0.13 to $0.15 per chip would lead to a decrease in demand
of about 1.5 15% 22.5%, or from 10,000 to about 7,750 chips per month.
Copyright © 2005. Stanford University Press. All rights reserved.
One of the appealing properties of elasticity is that, unlike slope, its value is independent of the units being used. Thus, the elasticity of electricity is the same whether the
quantity electricity is measured in kilowatts or megawatts and whether the units are dollars
or euros.
Example 3.4
Consider the bulk chemical whose price-response slope was estimated in Example 3.2. It showed a 20% decrease in demand (from 50,000 pounds to
40,000 pounds) from a 10% increase in price (from 10 cents to 11 cents). The
corresponding elasticity is 0.2/0.1 2 —an elastic response. What if the units
were euros and tons? It would still be a 20% decrease in demand (from 25 tons
to 20 tons) from a 10% increase in price.
Like slope, point elasticity is a local property of the price-response function. That is, elasticity can be specified between two different prices by Equation 3.4 and a point elasticity can
be defined by Equation 3.5. However, the term price elasticity is often used more broadly
and somewhat loosely. Thus, statements such as “gasoline has a price elasticity of 1.22” are
imprecise unless they specify both the time period of application and the reference price. In
practice, the term price elasticity is often used simply as a synonym for price sensitivity. Items
with “high price elasticity” have demand that is very sensitive to price while “low price
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
Created from apus on 2023-12-19 12:04:28.
Copyright © 2005. Stanford University Press. All rights reserved.
basic price optimization
45
elasticity” items have much lower sensitivity. Often, a good with a price elasticity greater than
1 is described as elastic, while one with an elasticity less than 1 is described as inelastic.
Elasticity depends on the time period under consideration, and, as with other aspects of
price response, we must be clear to specify the time frame we are talking about. For most
products, short-run elasticity is lower than long-run elasticity. The reason is that buyers have
more flexibility to adjust to higher prices in the long run. For example, the short-run elasticity for gasoline has been estimated to be 0.2, while the long-run elasticity has been estimated at 0.7. In the short run, the only options consumers have in response to high gas
prices are to take fewer trips and to use public transportation. But if gasoline prices stay
high, consumers will start buying higher mile-per-gallon cars, depressing overall demand
for gasoline even further. A retailer raising the price of milk by 20 cents may not see much
change in milk sales for the first week or so and conclude that the price elasticity of milk is
low. But he will likely see a much greater deterioration in demand over time. The reason
is that customers who come to shop for milk after the price rise will still buy milk, since it
is too much trouble to go to another store. But some customers will note the higher price
and switch stores the next time they shop.
On the other hand, the long-run price elasticity of many durable goods—such as automobiles and washing machines—is lower than the short-run elasticity. The reason is that
customers initially respond to a price rise by postponing the purchase of a new item. However, they will still purchase at some time in the future, so the long-run effect of the price
change is less than the short-run effect.
Finally, it is important to specify the level at which we are calculating elasticity. Market
elasticity measures total market response if all suppliers of a product increase their prices—
perhaps in response to a common cost change. Market elasticity is generally much lower
than the price-response elasticity faced by an individual supplier within the market. The
reason is simple: If all suppliers raise their price, the only alternative faced by customers is
to purchase a substitute product or to go without. On the other hand, if a single supplier
raises its price, its customers have the option of defecting to the competition.
Table 3.1 shows some elasticities that have been estimated for various goods and services.
Note that a staple such as salt is very inelastic— customers do not change the amount of salt
they purchase very much in response to market price changes. On the other hand, we would
expect that price elasticity of the market-response function faced by any individual seller of
salt to be quite large—since salt is a fungible commodity in a highly competitive market.
Table 3.1
Estimated price elasticities for various goods and services
Good
Salt
Airline travel
Tires
Restaurant meals
Automobiles
Chevrolets
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
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Short-run
elasticity
Long-run
elasticity
0.1
0.1
0.9
2.3
1.2
4.0
—
2.4
1.2
—
0.2
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basic price optimization
This effect can be seen in Table 3.1 in the difference between the short-run elasticity for automobile purchases (1.2) and the much larger elasticity (4.0) faced by Chevrolet models.
The table also illustrates the fact that long-run elasticity is greater than the short-run elasticity for airline travel (where customers respond to a price rise by changing travel plans in
the future and traveling less by plane), but the reverse is true for automobiles (where consumers respond to price rises by postponing purchases).
Copyright © 2005. Stanford University Press. All rights reserved.
3.1.3 Price Response and Willingness to Pay
So far, we have treated the price-response function as simply given. In reality, demand is the
result of thousands, perhaps millions, of individual buying decisions on the part of potential
customers. Each potential customer observes our price and decides whether or not to buy
our product. Those who do not buy our product may purchase from the competition, or
they may decide to do without. The price-response function specifies how many more of
those potential customers would buy if we lowered our price and how many current buyers
would not buy if we raised our price. Thus the price-response function is based on assumptions about customer behavior. We usually cannot directly track the thousands or even millions of individual decisions that ultimately manifest themselves in demand for our product.8 However, it is worthwhile to understand the assumptions about customer behavior that
underlie the price-response functions so that we can judge if the price-response function is
based on assumptions appropriate for the application. The most important of such models
of customer behavior is based on willingness to pay.
The willingness-to-pay approach assumes that each potential customer has a maximum
willingness to pay (sometimes called a reservation price) for a product or service. A customer
will purchase if and only if the price is less than her maximum willingness to pay. (We will
use willingness to pay, sometimes abbreviated w.t.p., to mean “maximum willingness to
pay.”) For example, a customer with a willingness to pay of $253 for an airline ticket from
New York to Miami will purchase the ticket if the price is less than or equal to $253 but not
if it is $253.01 or more. In this case, d(253) equals the number of customers whose maximum willingness to pay is at least $253. A customer with a maximum willingness to pay of
$0 (or less) will not buy at any price.
Define the function w(x) as the w.t.p. distribution across the population. Then, for any
values 0 p 1 p 2:
p2
w 1×2 dx fraction of the population that has w.t.p. between p and p
1
2
p1
We note that 0 w (x) 1 for all nonnegative values of x. Let D d(0), the maximum demand achievable. Then we can derive d(p) from the w.t.p. distribution from
d 1p2 D
w 1×2 dx
q
p
We can take the derivative of the corresponding price-response function to obtain
d¿1p2 Dw 1p2
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
Created from apus on 2023-12-19 12:04:28.
(3.6)
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47
which is nonpositive, as required by the downward-sloping demand curve property. Conversely, we can derive the willingness-to-pay distribution from the price-response function 9
using
w 1×2 d¿1x 2/d 102
Example 3.5
The total potential market for a spiral-bound notebook is D 20,000, and
willingness to pay is distributed uniformly between $0 and $10.00 as shown in
Figure 3.3. This means that
w 1×2 e
if 0 x $10
otherwise
1/10
0
We can apply Equation 3.6 to derive the corresponding price-response curve:
d 1p2 20,000
10
11/102 dx
p
20,00011 p /102
20,000 2,000p
The price-response curve d(p) 20,000 2,000p is a straight line with d(0)
20,000 and a satiating price of $10.00.
Example 3.5 illustrates a general principle:
w(p)
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A uniform willingness-to-pay distribution corresponds to a linear priceresponse function, and vice versa.
1/10
0
$10
Price
Figure 3.3 Uniform willingness-to-pay distribution.
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
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basic price optimization
One of the advantages of Equation 3.6 is that it partitions the price-response function
into a total-demand component D and a willingness-to-pay component w(x). This is often
a convenient way to model a market. For example, we might anticipate that total demand
varies seasonally for some product while the willingness-to-pay distribution remains
constant over time. Then, given a forecast of total demand D(t) for a future period t, our
expected price-response function in each future period would be
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d 1p 1t 2 2 D 1t 2
P
w 1×2 dx
(3.7)
p
This approach allows us to decompose the problem of forecasting total demand from the
problem of estimating price response. It also allows us to model influences on willingness
to pay and total demand independently and then to combine them. For example, we might
anticipate that a targeted advertising campaign will not increase the total population of potential customers, D(t), but that it will shift the willingness-to-pay distribution. On the
other hand, if we open a new retail outlet, we might anticipate that the total demand potential for the new outlet will be determined by the size of the population served, while the
willingness to pay will have the same distribution as existing stores serving populations with
similar demographics.
Of course, a customer’s willingness to pay changes with changing circumstances and
tastes. The maximum willingness to pay for a cold soft drink increases as the weather gets
warmer—a fact that the Coca-Cola company considered exploiting with vending machines
that changed prices with temperature (see Chapter 12 for a discussion of the “temperaturesensitive vending machine” idea). Willingness to pay to see a movie is higher for most
people on Friday night than on Tuesday afternoon. A sudden windfall or a big raise may increase an individual’s maximum willingness to pay for a new Mercedes Benz. To the extent
that such changes are random and uncorrelated, they will not effect the overall w.t.p. distribution, since increasing willingness to pay on one person’s part will tend to be balanced by
another’s decreasing willingness to pay. On the other hand, systematic changes across a population of customers will change the overall distribution and cause the price-response function to shift. Such systematic changes may be due to seasonal effects, changing fashion or
fads, or an overall rise in purchasing power for a segment of the population. These systematic changes need to be understood and incorporated into estimating price response and future price response.
A disadvantage of the willingness-to-pay formulation is that it assumes that customers
are considering only a single purchase. This is a reasonable assumption for relatively expensive and durable items. However, for many inexpensive or nondurable items, a reduction in price might cause some customers to buy multiple units. A significant price reduction on a washing machine will induce additional customers to buy a new washing machine,
but it is unlikely to induce many customers to purchase two. However, a deep discount on
socks may well induce customers to buy several pairs. This additional induced demand is
not easily incorporated in a willingness-to-pay framework—willingness-to-pay models are
most applicable to “big ticket” consumer items and industrial goods.
Phillips, R. (2005). Pricing and revenue optimization. Stanford University Press.
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49
3.1.4 Common Price-Response Functions
Linear price-response function. We have seen that a uniform distribution of willingness
to pay generates a linear price-response function. The general formula for the linear priceresponse function is
d 1p2 D mp
(3.8)
where D 0 and m 0. D d(0) is the demand at zero price. The general linear priceresponse function is shown in Figure 3.4. The satiating price—that is, the price at which demand drops to zero—is given by P D/m. The slope of the linear price-response function
is m for 0 p P and 0 for p P. The elasticity of the linear price-response function is
mp/(D mp), which ranges from 0 at p 0 and approaches infinity as p approaches P,
dropping again to 0 for p P.
Demand
D d(0)
d(p) D mp
P D/m
0
Price
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Figure 3.4 Linear price-response function.
We will use the linear price-response function in many examples because it is a convenient and easily tractable model of market response. However, it is not a realistic global representation of price response. The linear price-response function assumes that the change
in demand from a 10-cent increase in price will be the same, no matter what the base price
might be. This is unrealistic, especially when a competitor may be offering a close substitute. In this case, we would usually expect the effect of a price change to be greatest when
the base price is close to the competitor’s price.
Constant-elasticity price-response function. As the name implies, the constant-elasticity
price-respo
