Poway Unified School District and the Power of Compound Interest

In the past few days, Fox News and other organizations have reported about bonds that Poway Unified School District, near San Diego, CA, has issued.  The bonds were issued almost a year ago, in September 2011; their issue was approved by the school board almost two years ago (in October 2010).

The reports have centered around the assertion, or at least implication, that tax payers have been totally ripped off by the bond offering.  Here is a link to the Fox News article as it appeared on August 10, 2012:

California School District Will Spend $1 Billion to Borrow $100 Million

These bonds were issued after approval of voters of Proposition C on February 5, 2008.  The proposition that appeared on voters’ ballots read:

PROPOSITION C: “To provide safe and modern school facilities, improve student learning, and qualify for approximately $20 million in State matching money, shall School Facilities Improvement District No. 2007-1 of the Poway Unified School District issue $179 million in bonds at legal interest rates to upgrade aging classrooms, libraries, science & computer labs; replace roofs, plumbing, heating, ventilation and electrical systems; improve fire alarms and school security; remove hazardous materials; fund needed facilities, subject to mandatory audits, independent citizens’ oversight and without an estimated increase in tax rates?”

The proposition passed with 63.9% of the voters approving.

The article makes some strange assertions and presents many facts that are either incorrect or misleading.  This post discusses some of these inaccuracies and then discusses the financial aspects of the bonds.

Inaccurate Information in the Fox News Report

The Fox News report makes the following untrue statements:

  1. “Its being called a loan that not even a subprime lender would make.”  Who calls it that?  If the details of the loan make it such a raw deal for tax payers and such a good deal for banks, then why would a subprime lender not be interested?  Likewise, if the deal is such a bad investment for a lender, wouldn’t it be a good deal for the tax payers?  From either angle, this assertion makes no sense.
  2. “…they can’t be paid off early or refinanced.”  Yes, it’s true that non-callable bonds cannot be paid off early.  But, they can certainly be refinanced at the end of the term.
  3. “And the Poway district has already borrowed millions of dollars at nosebleed 12.6% interest rates.”  When?  In the 1980′s?  Poway’s CPA certified financial statements do not show any evidence of loans at such rates.  See  http://www.powayusd.com/depts/bss/finance/ for CPA certified audit reports of Poway’s budget, debts, etc.  In addition, even the lowest rated municipal bonds do not pay interest this high.  Remember that the market, not the school district, sets the interest rate.  The market sets the interest rate based on the credit worthiness of the school district.  Moody’s rates Poway’s credit as Aa2.  This is an investment grade rating.  In addition, Poway has had a Aa2 or better rating for at least the last 15 years.  Municipal bonds with such high credit ratings have never yielded interest rates at levels even close to 12.6% in the recent past.
  4. The article states that the underwriters of the bond issue will get $1.4 million in fees which, according to the article is a “sweet deal”.   The implication is that the underwriters are ripping off the tax payer.  In truth, these fees are less than 1.5% of the value of the bond offering.  While this amount may sound high, there is much more to underwriting a bond offering than meets the eye.  1.5% is within the range of “reasonable and customary” fees for bond underwriting.  This is not some sort of unusual “sweet deal”.
  5. The article states that school administration and teacher compensation chew up 85% of the school’s budget.  This is not true.  According to the certified financial statements from the school district, the district spent $244,737,036 on teacher compensation and administration in the 2010-2011 school year.   The total school budget was $337,800,038.  So, the real number is 72.45%.

It should be pointed out that the district budget for 2012 is $10 million less than it was for 2009 and that spending per pupil is less than $10000 per student, despite the fact that the San Diego area has a high cost of living.  $10000 per student is within the average range for primary and secondary public education cost in the United States.  Compare this to the many districts in New York that spend more than $20000 per student.

There are other inaccuracies and misrepesentations of facts in the article.  The reader of this blog is encouraged to dig into the financial reports of the district.

The True Financial Implications of Poway Unified School District’s Bond Offering

Interest on the $105 million bonds will accrue, but not be paid, for 20 years.  After 20 years, interest will be paid on the $105 million principle as well as the interest that will have built up over the first 20 years.  These types of bonds are called convertible zero coupon bonds.  The intricate workings of them can be difficult to understand.  However, it is easy to make an excellent approximation.

Over 40 years, Poway Unified School District will pay $981 million to pay off these bonds.  This is equivalent to the interest rate on a common bond or loan of approximately 5.75%.

Looking at this another way:  If you were to deposit $105 million into a bank CD promising to pay you 5.75% interest each year for the next 40 years, you would have just over $981 million in 2052!  This is the power of compound interest.

How does a 5.75% interest rate compare to other municipal bonds?  Well, as stated above, Poway Unified School District has a Aa2 credit rating.  This is not an uncommon credit rating for municipalities in the United States.  A quick check of today’s yields on 40 year municipal bonds with Aa2 credit ratings shows yields are right around 5% today.  Rates last fall for 40 year bonds were approximately 0.75% higher.  As expected, Poway paid the market rate based on its credit rating.

Closing Thoughts

Poway’s Unified School District’s credit worthiness is not any different than hundreds of municipalities around the country.  It is investment grade; meaning that buying its bonds is a fairly low risk investment.  The bonds were issued by the district with the approval of taxpayers.  The interest rate is at the market rate.

The bonds will not result in a tax increase for at least 20 years.  After 20 years, the interest paid will be a small percentage of the district’s budget.  In addition, inflation over the next 20 years significantly decreases the impact of the interest payments that will have to be made.  It is not unreasonable to envision a situation where minor budget cuts would completely offset the interest payments.

Instead of focusing on a fairly low risk scenario that is common to local governments throughout the country, and instead of misreporting a vast majority of the facts about this bond issue, news organizations could better serve the public by reporting on real dangers to future taxpayers.  One example of such a real danger is the chronic under funding of government pension plans, including Poway Unified School District’s teacher retirement plan.  Without significant tax increases in the future, its probable that drastic cuts in government services will be necessary for governments to meet their pension obligations.

Web Filtering – Limited Access Wireless

Imagine a business where employees need secure, full,  access to the internet.  At the same time, imagine that this business needs to provide customers and vendors with access. But, it must keep its proprietary information out of the hands of these visitors. And it needs to limit their access to websites with questionable content.

A similar need might arise in a home environment with children. Perhaps parents need full internet access while children only need access to websites that the parent approves of.

This post is the first of a series of posts that describe a method of accomplishing this.

The network system diagram shown below implements this system on a small-scale, as needed in a small office or home:

Open filtered internet and secured full internet simultaneously

Internet connectivity is provided by an ISP such as AT&T DSL, Time Warner Cable, or through a wireless internet provider such as Clear. Connection to the internet is through a modem.

An Ethernet cable connects the modem to a wireless router. The router shown above is a Linksys WRT54GL Router. The WRT54GL is certainly not the most sophisticated wireless router on the market. But, it is one of the most widely used wireless routers, has been on the market for more than 6 years, and has a track record of solid, reliable, performance. This router is configured to provide unfiltered access to the internet. Any computer or other WiFi device connected to it has full internet access. Communications between this WRT54GL and the devices connected to it are encrypted so that interception is very difficult.

The second Linksys WRT54GL, the one on the right, is connected to the first wireless router through an Ethernet Cable. It is configured so that it can be accessed without an encryption key or password. This means that data transferred through it is not secure and can be easily intercepted. However, data security is usually not desired for public internet hotspots. (If desired, encryption could be added to limit access).

This second wireless router filters the internet so that undesired websites are not accessible.

This basic system can be easily expanded by adding more wireless routers. An expanded system can provide:

  • Enterprise level wireless coverage for medium size businesses
  • Hotspot coverage over a larger area, such as an apartment complex, shopping mall, or outdoor area
  • Simultaneous filtered and non-filtered internet access
  • Multiple levels of filtering for different types of users

I am happy to help your organization with a custom or turn-key wireless system design.  However, subsequent posts will explain in detail how to set up these key components of a multi-privilege level WiFi system.

Should I Pay a Loan Off Early?

Dear Jim,

I have a 5 year loan for a vehicle that I took out 4 1/2 years ago.  I have 6 months  left to pay and the balance is just over $2400.  The interest rate is 7.75%.  I also have $3000 in cash that I am thinking about investing.  Should I invest it or should I pay off my car loan?

A friend told me that I’ve already paid most of the interest on the loan and that, from this point out, it’s not worth paying off since I’m pretty much just paying principle at this point.


Chuck, your friend is both right and wrong.

Assuming you haven’t made any extra payments along the way, it looks like you borrowed about $24000 to purchase your auto.  Over the past 54 months (4.5 years), you have paid about $5000 in interest.  But, if you keep the loan and finish making the payments over the next 6 months, you will only have to pay $65 in interest.  Your friend is absolutely correct that you have already paid the majority of interest on this loan.

However, it’s probably best to pay off the loan now.  Here are explanations of your two options:

  1. Pay the loan off now.  You will save $65 in interest.  Over the next 6 months, you can save ~$500 each month instead of making payments.  You’ll have $3000 in cash in 6 months that you can invest.  Even if you don’t get any interest on your savings over the next 6 months, you will have saved $65 in interest.
  2. Keep making payments and invest your $3000 in a safe investment like a CD.  You’ll pay $65 in interest on the loan.  If you look around carefully, you should be able to find a CD that pays 0.5% interest.  Bankrate.com can help you find the best CD rates.  But, at 0.5%, your CD will only provide you with $7.50 of interest income.  Depending on your tax situation, you could pay up to 26% tax (15% Federal plus 11% state if you live in Oregon or Hawaii) on this interest income.  At the end of 6 months, you will have $59. 45 less in your pocket.

It’s clear that the first option is usually better.

One exception may be in the case where you are among the 28% of Americans without emergency savings.  If you have no savings, it may be advisable from a psychological perspective to start building an emergency fund rather than saving the $65 by paying off the auto loan early.  You’ll still have $65 less at the end of 6 months.  But, often, people find it easier to borrow money rather than give up savings.  If you have savings, you’ll carefully think before you dip into it.  With no savings, however, it can be more tempting to label purchases as “emergencies” and charge them to a credit card.

No Longer Funding 401K?

On May 23, 2012, Yahoo Finance posted an article  by K. W. Callahan explaining why he is no longer funding his retirement account.  Here is a snapshot of Mr. Callahan’s article.


Mr. Callahan has a business degree from the Indiana University Kelley School of Business.  Unfortunately, he seems to have forgotten much of what he learned in business school.  There are many misconceptions about the way the equity markets work.  But, this post concentrates on the power of dollar cost averaging and how Mr. Callahan has missed out on a significant opportunity to grow his retirement savings.

Mr. Callahan explains that he stopped contributing to his 401K in late 2007.  At that time, his retirement account balance was about $38000.  He correctly points out that his balance was only $33000 in late 2011 – almost 5 years later.

Assuming he makes about $50000 per year, a 3% contribution would have amounted to $125 per month.  If he had saved $125 per month in a non-interest bearing account, he would have added $7500 to his savings over the course of 5 years.  That would have brought his retirement savings today up to around $40000.

But, a better strategy would have been to invest his 401K in a diversified basket of securities reflecting the total market value, such as the iShares Russell 3000 Index Fund (IWV), and keep contributing.  If Mr. Callahan had done this, he would have had around $45000 as of June 1, 2012.

The reason is simple.  With regular contributions at set intervals, investors are able take advantage of low points in the market.  For example, purchasing $125 could have bought 3.15 shares of  (IWV) on February 2, 2009 — at the low point of the market.  A similar investment of $125 on April 1, 2011,  would have purchased only 1.56 shares.  Catching the up’s and down’s of the market by purchasing set amounts at regular intervals allows investors to accumulate more shares when prices are low and less shares when prices are high.   This is dollar cost averaging.

Obviously, there were months that such a strategy would have left Mr. Callahan’s account with less than $38000.   But, over a reasonably long period of time (years, or decades),  dollar cost averaging using a diversified basket of securities has historically produced much better yields than buying securities all at once and then holding them.  And it has produced better results than government bonds, real estate, gold, or other investments.

Dollar Cost Averaging
Mr. Callahan's Investment Alternatives

The lesson is to keep investing a set amount regularly in a broad range of investments.  Never give up.


Canon MF4150 Printer in Ubuntu 9.04

Download the following file:


Open a terminal.

cd /tmp

sudo cp CNCUPSMF4100ZK.ppd /etc/cups/ppd

sudo cp pstoufr2cpca /usr/lib/cups/filter/

Open Firefox and go to the following URL:  http://localhost:631/

Click Add Printer

Enter a Name, Location, and Description.  These can be anything you like.  I made my name “MF4150″, location “HALL”, and Description “MF4150 Printer”.

Click Continue.

For Device, choose LDP/LPR Host or Printer.  Choose Continue.

Choose URI of lpd://

Choose Continue.

Under “Or Provide a PPD File” type in:  /etc/cups/ppd/CNCUPSMF4100ZK.ppd

Click Add Printer.

You may be asked for a username and password.  If so, use your root user name and password.

Finally, you should see a notice that the printer has been added successfully.  The screen will evenually refresh.

Click the “Home” button on the CUPS web configuration page.  Then, click the button that says “Manage Printers”.

In terminal:  sudo chmod 755 /var/spool/cups

Clock “Print Test Page”.

Economic Value Estimation

The first step to marketing is determining how products and services create value for customers. Value can vary between customers.

Value is the total savings, monetary gains, or satisfaction that a customer receives from using a product or service.

The difference between this value and what people actually pay is called “consumer surplus.”  People often won’t pay for the entire value of the product because they know that the price charged in the market is less.  The price charged in the market is called the economic value.

Reference Value is the price of the customer’s best alternative or substitute for a product or service.

Differentiation Value is the economic value placed on the difference between a product or service and its best substitute.  Remember that perceptions can be the cause of the differentiation value.  This often makes differentiation value hard to estimate since its difficult to measure the subjective value of unquantifiable benefits.

Total Economic Value
is the reference value plus the differentiation value.

Economic Value Estimation ®

Economic Value Estimation ® (EVE) is the process of measuring Total Economic Value. Graphically, the process looks like this:

(This picture was taken from http://mgmtblog.com).

The EVE for a product is not always what the consumer perceives the value to be. Some reasons are:

  • The consumer may not know about the differentiating features of the product and doesn’t want to spend the time to find out.
  • The brand image may convince the consumer that some product is worth more than the EVE.
  • The customer may not be too picky about getting the most for her money.

Therefore, EVE only provides a good estimation of the maximum amount that can be charged for a product, assuming that consumers recognize all of the value that the product provides. If a product is under priced compared to its EVE, a good solution is to maintain or slightly raise prices while engaging in an aggressive consumer education campaign.

EVE is not the same for each consumer. That’s why price segmentation is beneficial, if possible.

The relationship between prices and economic value delivered can differ depending on circumstances. For example, new products usually must be priced below economic value to encourage people to try them. The opposite may be true for existing products with existing customers since consumers may not be motivated to switch. Or, perhaps, consumers don’t even care too much about determining economic value. This attitude of indifference toward price leads to a price premium called a regulation premium.

Its important for a firm’s sales force to use EVE so that they and the are focused on value of each part of an offering rather than on “price.” When there is price resistance, EVE can be used to present the customer with a less expensive alternative with fewer features. EVE can also be used to price segment among groups of consumers who need different bundles of features.

There are no shortcuts for estimating EVE. The steps for estimating EVE are:

  1. Study Customer Economics – Study customers’ objectives and their “next best competitive alternative.” The next best competitive alternative’s price is the reference price.
  2. Quantify Value Drivers – Customer depth interviews are the best source of information. The most important piece of information to try to obtain is the value driver algorithms. What does the consumer’s economic model say about the kinds of things that drive value?Note that its critical to do a good approximation about what drives consumer value rather than trying to do a complex calculation yielding an exact, but meaningless, number.
  3. Estimate Differentiation Value – Estimate the impact of the product on the marketplace by estimating the monetary value that consumers give to it above and beyond the value they give to the reference product. Remember that this can be positive or negative since a particular product is seldom better in all respects.Needless to say, the sum of all of the differentiation attributes must be positive in order for a product to sell at a higher price than a competing product. When trying to determine differentiation value, be sure to use equivalent units when asking the consumers to compare a product with a reference product. For example, two units of a firm’s product may replace three units of the reference product. It is important for consumers to be aware of this when their attitudes are surveyed.Use the estimate of differentiation value and the quality value drivers to build the economic value model.

A common problem in marketing communications is documenting the economic value. For example, an advertiser might look at the total circulation of a publication and purchase advertising based on cost per customer reached. In reality, she should be looking at the cost of reaching the kinds of customers that she really wants to reach, how long they spend reading the publication, etc. To convince the advertiser to purchase advertising space, the publication may have to learn a significant amount about the advertiser’s business so that the price differential between its prices and those of other magazines can be justified.

Commodity products are essentially identical products that are offered by more than one manufacturer. A good example is a cheap portable radio. There is no differentiation between products. Since there is perfect competition in commodity markets, prices will be set very close to the actual economic value of the product.

Consumer Value Modeling (CVM)

CVM relies on customers’ subjective judgments about price and attribute based performance. CVM assumes that consumers look for products that give them the most benefit for the money and time invested in using it. It rates various product strengths or weaknesses according to the weight that the consumer puts on them. Then, CVM tries to create an average linear relationship between price and perceived value for the attributes.

Consumer Value Modeling emerged from the TQM movement. Companies tried to use it to measure and deliver superior quality at a competitive price. The problem is that people may be willing to pay for a particular attribute of a product or may be willing to pay for combinations of attributes. Each consumer is different.

The biggest problem with CVM is that it underestimates the value of significantly different products for people who are willing to pay premiums for combinations of attributes. On the other hand, it also overestimates the value of products that are similar to each other.

Simple Price Concepts

All firms set prices for their products. Poor pricing is easy. But, pricing well is very difficult. Proper determination of product pricing requires knowledge, preparation, and insight. Good pricing separates top executives from the pack.

The four P’s of marketing are:

  1. Product
  2. Price
  3. Placement
  4. Promotion

Pricing is the most important. The reason is that the other 3 P’s all deal with creating value whereas pricing is method for capturing value. Bad pricing decisions decrease the effectiveness of product, placement, and promotion.

A firm’s environment costs of:

  1. Costs
  2. Demand
  3. Competition
  4. Antitrust/Legal

Slotting Allowance

Slotting Allowance is money that a firm gives to a retailer for the privilege of putting a product on the store shelf. The amount of money spent on slotting allowances alone is more than all of advertising and consumer promotion expenditures, such as coupons, combined.

Smart Cards

Many stores, especially grocery stores, use smart cards to track a customer’s purchasing behavior. A customer is encouraged to scan her cards each time she makes a purchase. The retailers collect purchasing data into a database with her name, all of the purchases she’s ever made, and what kinds of coupons she uses. This information can be used to determine price sensitivity and can be used to create targeted pricing schemes. In theory, the data can be used to charge different prices to different individuals for the same product. One supermarket chain in California is doing this. There is a kiosk in supermarket which displays customized special offers when the smart card is inserted. Another retailer has a terminal display on the shopping cart itself where a customer puts your card in the slot. The cart buzzes if it wants her attention. For example, it might tell her that she will get 25 cents off the cookies she just passed.

Profit Maximization – Raise Prices or Increase Market Share?

Consider the following situation:

A soap manufacturer has annual sales of 100000 units. The selling price is $1 per bar. It costs $30000 to set up a factory to make soap, and it costs $0.60 to make each bar.

$3000 are fixed costs and $0.60 are variable costs. Fixed costs do not change as more bars are made. But, variable costs increase each time the firm makes another bar of soap. The profit of the firm is:


Profit = (100000 bars)*($1 per bar) – $3000 – (100000 bars)*($0.60 per bar)


= $100000 – $30000 – $60000

= $10000

Would the firm be better off if sales were increased by 1% (1000 bars) or if, instead, the price were increased by 1% ($0.01)?

Profit for 1% Increase in Sales = (101000 bars)*($1 per bar) – $3000 – (101000 bars)*($0.60 per bar)

= $101000 – $30000 – $60600

= $10400

Profit for 1% Increase in Price = (100000 bars)*($1.01 per bar) – $3000 – (10000 bars)*($0.60 per bar)

= $101000 – $30000 – $60000

= $11000

This example can be extended to firms in general. A study by McKinsey and Company based on average economics showed that, on average, a 1% increase in a product’s price cases a 11.1% increase in profits. But, a 1% increase in sales, on average, only results in a 3.3% increase in the firm’s profit.

Some Easy, but Poor Ways to Set Price

Firms often use these easy techniques for setting price; but they result in foregone profits:

  • Cost-Driven Pricing

  • Customer-Driven Pricing

  • Competition-Driven Pricing

Cost-Driven Pricing

Firms engaging in cost-driven pricing set prices based on their costs and profit objectives. Cost-Driven Pricing is sometimes called “Cost Plus Pricing.”

A firm typically puts a significant amount of money upfront into product design. Then, the engineering team turns the product design over to the manufacturing team which determines how to produce the product at a cost that will be at least marginally profitable. Next, the finance team is asked for determine, based on per unit cost and markup, what the product selling price should be. Finally, the marketing and sales group is tasked with sell the product at the cost that has been predetermined. Here is an Example of Cost-Driven pricing:

A firm has a product which has a marginal cost of $3 per unit. This marginal cost is constant regardless of the volume produced. The fixed costs for setting and remaining in production are $4.5M.

The per unit cost is $3 per unit plus ($4.5M / [Number of Units Produced]). Obviously, the per unit cost decreases as the number of units produced increases.

Many companies get around the lack of a static unit cost by artificially setting a desired production rate. So, let’s assume that this firm decides it wants to sell a million units. Then, the price per unit is:

Price Per Unit = $3 + ($4.5M/ 1M) = $7.50

The firm wants to make a profit of $1.50 per unit. So, it will try to sell the product for $9.

But… Will customers really pay $9 for the product? What are competitors charging? The sales number of one million units is totally made up!

What if the firm is only able to sell 750000 units at $9 each? It will only break even. That’s obviously not the desired outcome. Does the firm raise price? Lower the price? It would need to know what the demand curve for the product looks like in order to answer these questions. Otherwise, there’s no way of knowing what the effect of changing the price will be.

The problem is that Cost-Driven Pricing gives an apparent sense of objectivity to the price. The price is not objective.

A Cost-Driven Pricing scheme ignores customers. It fails to capture customers’ additional willingness to pay. The firm has no idea about what customers are willing to pay – if they’re willing to pay anything at all!

Another problem with Cost-Driven Pricing is that it ignores competitors. Competitors could be charging a lot less for similar items.

Cost-Driven Pricing usually results in charging one price to all customers. A one price fits all philosophy is often not a good practice because the firm cannot engage in price discrimination.

There are advantages to this pricing scheme. Its easy to understand and implement. If it works, the firm will have cover all of its variable costs and, in addition, the markup should cover some of the business’ general administrative costs. A decent return on investment will result.

These disadvantages of Cost-Driven Pricing outweigh the advantages.

Customer Driven Pricing

The problems with Cost Driven Pricing can be avoided by asking what price that customer is willing to pay. Then, the firm charges the maximum price that the customer is willing to pay.

There are problems with Customer Driven Pricing too. First, Customer Driven Pricing ignores the competition. And it ignores the costs of making the product.

Customer Driven Pricing is particularly poor for pricing a new product because customers don’t know how much they like it or how much value they place on it. Lastly, if customers know this is how a firm is setting prices, why would they tell the firm what they’re willing to pay? They have no incentive to tell the truth.

Competition Driven Pricing

Competition Driven Pricing focuses on what the competition is charging.

Problems with Competition Driven Pricing:

  • Price wars can result.
  • Competition Driven Pricing ignores costs. A competitor may have a whole different cost structure which allows them to make a profit at a much lower selling price.
  • Ignores customer willingness to pay. The firm’s product may be better or worse than the competitors, making the customer willing to pay more or less for it.

Each of these strategies as positive points. But, none of them are optimal. Optimal pricing strategy takes in account costs, customers, competition, and the legal environment.

Price Elasticity of Demand

The demand for a product varies depending on the price charged.  For some products, demand changes only slightly with a large increase in price.  For other products, demand changes significantly with only a small increase in price.  The Price Elasticity of Demand is defined as follows:

  Price Elasticity of Demand = (∂Q / ∂P) * (P / Q)

Where Q is quantity demanded, P is price,  ?Q is change in quantity demanded, and ?P is change in price.

So, supposed that a family demands 20 gallons of gas per week when the price is $1.99 per gallon.  But, if the price is $2.02 per gallon, the family only demands 19.5 gallons of gas.  The price elasticity is:

 Price Elasticity of Demand for Gas = (0.5 / 0.03) * (1.99 / 20) = 1.66

The meaning of this value is that a 1% increase in price of gasoline reduces the family’s demand for gasoline by 1.66%.

Of course, price elasticity will depend on what kind of product is sold (i.e., whether its a necessity or a luxury item), the income level of the consumers, and will also depend on the locality.  For example, in developing countries, the price elasticity of demand for gasoline will probably be much larger than it is in the developed world.  Also, a product with many close substitutes is more likely to have a larger elasticity.

We can see from the above equation that a price elasticity of demand close to zero means that demand will not change significantly with price.  But, a large value indicates that the demand changes significantly with price.  Such a large price elasticity is often called a “flat demand curve.”  If the demand curve is completely flat, an unlimited quantity can be sold at a given price; but, nothing will be sold if the price is raised only slightly.

Its important to note that the price elasticity of demand is a snapshot at a certain point on the demand curve.  The 1.66% reduction in demand for a 1% increase in price of gasoline may be true for the family when gasoline is priced around $2 per gallon, but will probably not be true when the price of gasoline is $4 per gallon.  Perhaps, at $4 per gallon, the family will reduce its demand for gasoline by 3% for every 1% increase in the price of gas.  Keep this in mind when calculating the price elasticity of demand with data with large differences in price.  If such varying data is the only data available, a more accurate way to calculate the elasticity may be to average the price and demand over the range resulting in the following equation:

Price Elasticity of Demand = (ΔQ * [P1 + P2])  /  ( ΔP * [Q1 + Q2])

Price Elasticity of Demand from the Demand Curve

Suppose that it is know that a product has the following demand linear curve:

P = a – bQ

Where a and b are constants.   It can be shown using the above equations that:

Price Elasticity of Demand =  (b*P)/Q

With this linear demand curve, the price elasticity of demand approaches zero as the price approaches 0.  This makes sense if we again consider gasoline as an example.  A family will likely acquire the same amount of gasoline if the price is 1 cent per gallon as it would acquire if the price were 2 cents per gallon.

Likewise, as Q approaches 0, the price elasticity of demand approaches infinity.  Using the gasoline example:  If a family demands one gallon of gas per week if the price is $20 per gallon, a 10% increase of price to $22 per gallon might cause it to cut its consumption to half gallon per week.

Price Elasticity and Revenue Maximization

For a firm, the important question to ask is:  What price should be charged to maximize revenue?

The total amount spent by customers on a product is P*Q.   This is the firm’s total revenue.  In the simple case where price is = a – bQ:

TR = P*Q = (a-bQ)Q = aQ – bQ2

A price is considered elastic if the price elasticity of demand is > 1.  The total amount spent by customers decreases when price rises.

A price is considered inelastic if the price elasticity of demand is < 1.   So, the total amount spent by customers increases when price rises.

The equilibrium point occurs when the price elasticity of demand equals 1.  This is the price point which maximizes total revenue.

Marginal Revenue

We know that total revenue is described by the equation:

TR = P*Q = (a-bQ)Q = aQ – bQ2

The total revenue increases as the quantity increases up to a certain point and then it begins to fall.   The amount of revenue (or cost) of the final item produced is called the Marginal Revenue.  The Marginal Revenue is the first derivative of the Total Revenue:

MR = a – 2bQ

Note that the marginal revenue curve has a slope that is exactly twice the slope of the demand curve.

If marginal revenue is positive, then making extra units of the product will increase the total revenue.  If it is negative, then making extra units will decrease the total revenue.  Obviously, once marginal revenue comes equal to zero, no more units should be produced.  This is the point where price elasticity is equal to 1.

Another way to write the equation for marginal revenue is:

MR = P * (1 -[1 / Price Elasticity])

Maximizing Profit

Maximizing profit and maximizing revenue are not the same thing.  If the marginal cost of making one extra unit of a new product is $2 and the marginal revenue obtained by selling that product is $1,  producing it does not make sense.  The profit maximizing point is where marginal cost equals marginal revenue:


MC = P * (1 -[1 / Price Elasticity])

For example, if the marginal cost to product a product is $10 and the price elasticity of demand is 2,

10 = P * (1 – [1/2])

P = $20 should be the selling price

Note that, as the price elasticity increases, the optimal selling price decreases.  So, if it costs $10 to produce a product and the price elasticity of demand is 4:

10 = P * (1 – [1/4])

P = $13.33 should be the selling price

Obviously, determining the price elasticity of demand is critical for a firm when setting the price of a product.