Business

Inflation: How long does it take for prices to double?

A question that often comes up is how long it would take for prices to double if the rate of inflation remained constant. It also helps to turn an abstract percentage number into a value that is easier to grasp and interpret.

If we start at a certain value for the consumer price index CPI0 and apply a constant annual inflation factor f (which is just the annual inflation rate expressed in decimals plus one), the CPI would grow exponentially according to this formula:

CPIn = CPI0 · f n

where CPIn symbolizes the Consumer Price Index for year n. The prices have doubled when CPIn equals 2 · CPI0. So we get:

2 · CPI0 = CPI0 · f n

Or, after solving this equation for n:

n = ln(2) / ln(f)

with ln being the natural logarithm. Using this formula, we can calculate how many years it would take for prices to double given a constant inflation rate (and thus inflation factor). Let’s look at some examples.

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In 1918, the end of World War I and the beginning of the Spanish Flu, the inflation rate in the US rose to a frightening r = 0.204 = 20.4 %. The corresponding inflation factor is f = 1.204. How long would it take for prices to double if it remained constant?

Applying the formula, we get:

n = ln(2) / ln(1.204) = ca. 4 years

More typical values for the annual inflation rate are in the region several percent. Let’s see how long it takes for prices to double under normal circumstances. We will use r = 0.025 = 2.5 % for the constant inflation rate.

n = ln(2) / ln(1.025) = ca. 28 years

Which is approximately one generation.

One of the highest inflation rates ever measured occurred during the Hyperinflation in the Weimar Republic, a democratic ancestor of the Federal Republic of Germany. The monthly (!) inflation rate reached a fantastical value of r = 295 = 29500 %. To grasp this, it is certainly helpful to express it in form of the doubling time.

n = ln(2) / ln(296) = ca. 0.12 months = ca. 4 days

Note that since we used the monthly inflation rate as the input, we got the result in months as well. Even worse was the inflation at the beginning of the nineties in Yugoslavia, with a daily (!) inflation rate of r = 0.65 = 65 %, meaning prices doubled every 33 hours.

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This was an excerpt from “Business Math Basics – Practical and Simple”. I hope you enjoyed it. For more on inflation check out my post about the Time Value of Money.

The Service Recovery Paradox – Customer Service At Its Best

Let’s say you buy a coffee machine. You go home, plug it in and it does what it’s supposed to do – make coffee. Depending on how easy it is to use, how well the coffee tastes, etc … you experience a certain level of satisfaction. Now suppose one day the coffee machine suddenly stops working. Obviously, your satisfaction will drop sharply. You call customer service and after a lot of try this and try that, all eating up your precious time, it works again. Your satisfaction will rise again, but not to the initial level. You might even decide that next time you’ll not buy from this company again, they don’t seem to be able to provide functioning machines.

This is how the story usually goes. But there’s a very interesting paradox that can lead to a surprising outcome. Given certain conditions (see below) and an exceptional customer service, studies have shown that after the failure the level of satisfaction can rise above the initial level. In other words: customers who have experienced a problem with the product and have been successfully helped by the manufacturer’s customer service can be more satisfied with the company than those customers who have not experienced any problem at all. This is called the service recovery paradox. A widely cited work regarding this paradox by Hart et al. (1990) in the Harvard Business Review states: “A good recovery can turn angry, frustrated customers into loyal ones. It can, in fact, create more goodwill than if things had gone smoothly in the first place”.

I made a crude graph to visualize this situation. Note that in a standard recovery the level of satisfaction rises, but not beyond the initial value. This is the situation we usually experience. A paradoxical recovery propels the level of satisfaction past this initial value.

Image

There are some conditions that need to be met in order for the paradox to be able to occur.

The effect of the severity of the failure

 According to McCollough et al. (2000), satisfaction varies with the severity of the failure. Many service problems that customers experience are only mildly annoying, while others can be very severe. Hoffman et al. (1995) state that the higher the severity of the failure, the lower the level of customer satisfaction. Consequently, the existence of a recovery paradox depends on the magnitude of the failure. For example, perhaps an apology, empathy, and compensation could create a paradoxical satisfaction increase after a 20-minute wait at the front desk of a hotel. But would this paradoxical increase occur if the wait caused the guest to miss a flight? It is unlikely that any realistic recovery is capable of completely erasing the harm caused by such a severe failure.

In the event of a service failure, a recovery paradox is more likely to occur if the service failure is less severe than if the failure is more severe.

The effect of a prior failure

 A person’s satisfaction is a cumulative evaluation of all experiences with the firm (Cronin and Taylor, 1994). If the service failure occurred in a one-time only use, then the satisfaction judgment would be transaction-specific. However, an individual generally has a history of interactions with the firm, in which case satisfaction reflects the cumulative interactions over time between the individual and that firm (Bitner and Hubbert, 1994; Crosby and Stephens, 1987).

In the event of a service failure, a recovery paradox is more likely to occur if it is the firm’s first failure with the customer.

 The effect of the cause of the failure

 Service failures with persistant causes are more likely to repeat than failures with temporary causes. For example, when a hotel guest is assigned to an incorrect room category due to an outdated computer system, this could be considered a failure with a persistent cause. On the other hand, if the guest’s room assignment was botched because the front desk associate is in the initial stages of training, this could be viewed as an temporary cause. Customers are likely to be more forgiving of failures with temporary causes (Kelley et al., 1993). This is because the likelihood of a future inconvenience is minimal. Thus:

In the event of a service failure, a recovery paradox is more likely to occur if the customer perceives that the failure had a temporary cause.

The effect of perceived control

A service failure is any situation where something goes wrong, irrespective of responsibility (Palmer et al., 2000). Nevertheless, “the perceived reason for a product’s failure influences how a consumer responds” (Folkes, 1984, p. 398). Customers are more forgiving if they perceive that the firm had little control over the occurrence of the failure (Maxham and Netemeyer, 2002). Conversely, customers are less forgiving when they feel that the failure was foreseeable and should have been prevented (Folkes, 1984). For instance, did a wait occur because of a random spike in demand, or did it occur because the firm did a poor job in forecasting, planning or staffing? A bank customer may be understanding of a wait inside a bank lobby if there is an unexpected inflow of customers during a typically slow hour. On the other hand, the same customer may be less understanding if there is only one teller working during lunch hour on a Friday afternoon. Thus:

In the event of a service failure, a recovery paradox is more likely to occur if the customer perceives that the firm had little control over the cause of the failure.

For more on customer service, check out the 7 Laws of Customer Service.

The 7 Laws of Customer Service

If you want to provide solid and helpful customer service, you should stick to a number of basic rules that make you stand out from the crowd. Here are 7 rules that can help in doing so:

1. Roll Out The Red Carpet For Everyone. If there’s one thing people hate about poor service, it’s getting treated differently from others. It makes them feel inferior and second-class. Gary Richter says you should roll out the red carpet for everyone, but particularly those who don’t expect it. “I tell my employees, if we roll out the red carpet for a billionaire, they won’t even notice. If we roll it out for millionaires, they expect it. If we roll it out for thousandaires, they appreciate it. And, if we roll out the red carpet for hundredaires, they’ll tell everyone they know.”

2. Take Time To Know Your Customers. The fast pace of modern living together with advances in technology have together put a non-human face on much of our customer service. If you can find a way to re-connect with your customers one-on-one, you’ll strike a chord with your customers that will be like a streak of gold. Kathy Burns remembers a time when people took time to care and listen. “Some of you may remember, and others may have heard stories about, a time in life when the doctor would come to your home to check on you if you were ill. Or maybe you’ve heard about going down to your local pharmacy and having the owner greet you by name and ask how you’re doing. Not only did they ask, but they really wanted to know the answer and they took the time to listen to what you had to say. That’s customer service – taking the time to know your customers, really caring about how they feel, and wanting to go the extra mile to make sure they’re happy.”

3. Be Easy To Do Business With. One of the problems with modern businesses is that the systems we use to save time and money are often devised for the company’s benefit and not the customers. As a result, the customer experience is frustrating and difficult. Tracey Lowrance says this needs to be reversed. “Customers expect single source service. Customers don’t want to be transferred to every unit of your business to have their problems solved. They want to be able to do business with you with the slightest amount of discomfort. You must be easy to do business with.”

4. Go Out Of Your Way To Make Sure They’re Happy. One of the most important things your customers want from you is a guarantee that your product or service will work. So move heaven and earth to make sure it does. Bob Leduc suggests you shouldn’t make people pay until they are fully happy. “Instead of offering a money back guarantee, a service business can provide a guarantee to solve the customer’s problem. For example, a plumber can guarantee to come back without charge as often as necessary to stop the leak. A landscaper can replace without charge any plants that don’t survive for at least 6 months. A sales consultant can continue working without charge until the promised sales results are achieved.”

5. Notice What Customers See. A big part of what customers think about you comes from what they see and believe. Personal Selling Power noticed the following difference in two candy stores. “Although two competing candy stores had the same prices, neighbourhood kids preferred one store to the other. When asked why, they said, “Because the person in the good store always gives us more candy. The girl in the other store takes candy away.” True? Not really. In the good store the owner would always make sure to put a small amount of candy on the scale and then keep adding to it. In the bad store, the owner would pile a heaping amount of candy on the scale, and then take it off until it hit the right weight. The same amount of candy was sold, but perception is everything.”

6. Work On Everything The Customer Experiences. The customer experience isn’t just receiving the service or buying the goods. It’s about all the other little bits and pieces in-between. Such as the manner of the receptionist, the state of the floors and tables, the attitude of other staff, the ease of parking, the tone of the notices, the smile or lack of it on the face of the checkout team. Be like the Mirage hotel in Las Vegas who have a slogan that says: “We spend 600 hours a week pampering the plants. Imagine what we’ll do for our guests.”

7. Believe In Customer Service. To become a great service organization, you have to believe in customer service from the bottom of your soul. It has to be part of the way you work. Anita Roddick, founder of retail cosmetic franchise group Body Shop puts it like this: “I am still looking for the modern equivalent of those Quakers who ran successful businesses, made money because they offered honest products and treated people decently, worked hard, spent honestly, saved honestly, gave honest value for money, put back more than they took out and told no lies. This business creed, sadly, seems long forgotten.”

If you’re interested in learning more about customer service, be sure to check out the Service Recovery Paradox.

Computing and Tracking the Amazon Sales Rank

The webpage http://www.novelrank.com/ provides a very neat simple way to track the sales rank of any book on Amazon. This service is completely free.

The sales rank is computed from the sales rate. The more a book sells per day, the lower the rank will be.  Here’s an approximate formula, taken from: http://www.edwardwrobertson.com/2013/02/a-quick-way-to-calculate-amazon-sales.html.

100,000 / rank = sales per day

So if a book is on rank 50,000, it sells about twice a day. As far as I know, a borrow counts as a sale and a free download as one third of a sale.

I use novelrank to track my ebooks. This is what the output looks like (launch of “Great Formulas Explained”):

novelrankgreatformulas

Indeed a neat tool to see how a book is performing. Note that the tracking starts on the day you add it, dates before that are not shown.

As you can see, during the period when no sale is made the sales rank increases more or less linearly at about # 50,000 per day. The average rank during this time can be calculated by the formula: final minus initial rank divided by 2. When a sale is made, the rank makes a discontinuous jump to a lower value.

The Time Value of Money and Inflation

To make a point, I’ll start this blog entry in an unusual way, that is, by talking about vectors. A vector is basically an ordered row of numbers. Consider this expression for example:

(12, 3, 5)

This vector could represent a lot of things. For example a point in a three dimensional coordinate system, with the vector components being the x-, y- and z-values respectively. Or for a company offering three products, it could stand for the sales of these products in a certain year.

Why this talk about vectors? You were probably very surprised when you heard grandma say that she paid only 150 $ for her first car. It seems so amazingly cheap. But it is not. Your dear grandma is talking about 1950’s money, while you are thinking of today’s money. These two have a very different value.

If you want to specify the costs of a good precisely, merely giving an amount of money will not be sufficient. The value of money changes over time and thus to be absolutely precise, you should always couple this amount with a certain year. For example, this is what grandma’s car really cost:

(150 $, 1950)

This is far from (150 $, 2012), which is what you were thinking of when grandma shared the story with you. Using an online inflation calculator, we can conclude that this is actually what the car would cost in today’s money:

(1410 $, 2012)

Not an expensive car, but certainly more than 150 $ in today’s money. Now you can see why I started this chapter using vectors. They allow us to easily and clearly couple an amount with a year. A true pedant would even ask for one more component since we are still missing the respective months. But let’s not get too pedantic.

How can we justify saying that 150 $ in 1950’s money is the same as 1410 $ in today’s money? We can look at how much of a certain good these amounts would buy in the given year. With 150 $ in 1950 you could fill your basket with about as many apples as you can with 1410 $ today. The same goes for most other common goods: oranges, potatoes, water, cinema tickets, and so on.

This is inflation, goods get more expensive each year. At a later point we will take a look at what reasons there are for inflation to occur. But before that, let’s define the rate of inflation and see how it is measured …

This was an excerpt from the ebook “Business Math Basics – Practical and Simple”, available for Kindle here: http://www.amazon.com/dp/B00FXB8QSO.

Typical Per-Page-Prices for Ebooks

I did a little analysis of ebook prices per 100 pages for different categories in the Amazon Kindle store. In each category I looked at the top 12 paid books. This data can help readers to judge prices and authors to set them. Here are the results in increasing order:

Erotica: 1.7 $ per 100 pages (ranging from 1.0 – 3.1 $ per 100 pages)
Sci-Fi and Fantasy: 1.8 $ per 100 pages (ranging from 0.8 – 4.4 $ per 100 pages)
Short Stories: 2.0 $ per 100 pages (ranging from 0.5 – 4.2 $ per 100 pages)
Self-Help: 3.6 $ per 100 pages (ranging from 1.3 – 6.7 $ per 100 pages)
Applied Math: 4.0 $ per 100 pages (ranging from 0.9 – 7.9 $ per 100 pages)
Economy / Business: 7.2 $ per 100 pages (ranging from 3.3 – 17.2 $ per 100 pages)

Typical (and in my opinion fair) prices seem to be 2 $ per 100 pages for fiction and 4 $ per 100 pages for non-fiction. In the special case of business books, prices of 7 $ per 100 pages seem common.

From Simple to Compound Interest

Imagine you loan a bank the principal P = 10000 $ at an interest rate of i = 5 %. This is the amount of interest you would receive with simple interest, given the duration t of the loan:

t = 1 year
→ I = 10000 $ * 0.05 * 1 = 500 $

t = 2 years
→ I = 10000 $ * 0.05 * 2 = 1000 $

t = 3 years
→ I = 10000 $ * 0.05 * 3 = 1500 $

As you can see, the interest grows linearly with the duration of the loan. For each additional year, you get an additional 500 $, which is just 5 % of the principal 10000 $. In other words: each year the interest rate is applied to the principal. How could that be any different?

Consider this: At the end of the first year, you’ll receive an interest payment in the amount of 500 $. This means that your bank statement will now read 10000 $ + 500 $  = 10500 $. So why not apply the interest rate to this updated value? This would lead to an interest payment of 10500 $ * 0.05 = 525 $ for the second year instead of just 500 $.

Continuing this train of thought, at the end of the second year your bank statement would read 10000 $ + 500 $ + 525 $ = 11025 $. Again we would rather have the interest rate applied to this updated value instead of the unchanging principal. This would result in an interest payment of 11025 $ * 0.05 = 551.25 $ for the third year.

For comparison, here’s what the final pay out would be for the simple interest plan:

10000 $ + 500 $ + 500 $ + 500 $ = 11500 $

And this is what we would get with the “not simple” interest plan, where we apply the interest rate to the updated amounts instead of the principal:

10000 $ + 500 $ + 525 $ + 551.25 $ = 11576.25 $

The latter is called compound interest. It means that we include already paid interests in the calculation of next year’s interest, which leads to the amount received growing exponentially instead of linearly.

(This was an excerpt from “Business Math Basics – Practical and Simple”. You can get it here: http://www.amazon.com/Business-Math-Basics-Practical-Simple-ebook/dp/B00FXB8QSO/)