Appendix        Elasticity Policy Cases and a New Calculation Method

In Chapter 3, we present arguments for both an elastic demand and an inelastic demand for luxury goods.  On the one hand, rich people do not have to be as concerned with their budgets as poorer people do.  Thus, they may respond to an increase in the price of a luxury good by purchasing nearly the same amount at the higher price. On the other hand, the demand for luxury goods may be price elastic, compared to so-called necessities, since postponing their purchase causes little hardship for the buyers.  Which effect dominates, in practice?

In 1990, the democratic-controlled Congress of the US passed a luxury tax on yachts, furs, jewelry, private jets, and luxury cars.  Their logic was twofold: hit the rich with a greater share of the tax burden, and generate plenty of new revenue, since demand for such luxury items is price inelastic.  Taxes on inelastic items are the most efficient way to raise revenues, since little distortion of consumption takes place.  In other words, with very inelastic demand (picture a very steep demand curve), there is little change in consumption in response to the higher price, thus the deadweight social loss triangle associated with the distortion has a very small base, and a small area. See the illustrations below, and the detailed discussion of the concept of deadweight social loss in Chapter 7. 

Many products with inelastic demand, which are heavily taxed at present, have a regressive impact, i.e., poor people pay a disproportionate share of the tax burden.  Poor people spend a greater fraction of their income on cigarettes, liquor, and gasoline than do the rich.  They pay a greater share of those “sin” taxes.  (Is driving a car to work alone really a sin?  Well, it certainly isn’t a virtue!). 

Congress believed this luxury tax represented a chance to hit the rich for at least their fair share, a move likely popular with their core constituencies, including organized labor. Congress was dead wrong! Demand for luxury items was much more elastic that they had believed, for several important reasons.  First, there are many ways to spend an extra $100,000 or so to increase happiness, besides buying a yacht or Rolls Royce.  Secondly, a yacht, in particular can be purchased off-shore and tax-free in the South Pacific or Caribbean Islands , which is where many rich people do their yachting anyway.  Sales of yachts in Florida fell by almost 90 percent.  Congress had not considered that yachts purchased abroad could be close substitutes for yachts purchased domestically.  Thirdly, many rich people treated the tax as a temporary price hike, and delayed purchases while lobbying behind the scenes to repeal the tax. 

Congress soon faced another problem.  The buyers of yachts and private planes may be rich, but the workers who make them are part of the Democratic Party’s core constituency.  As orders dropped sharply, so did employment of unionized production workers in the factories for some of these luxury goods.  Complaints from these workers were the last straw for Congress, which already faced revenues well below projections ($1.5 billion dollars of revenue was expected over the first five years of the tax) and pressure from rich campaign contributors.  The tax was repealed after less than a year.  Note that purchases of some of these luxuries surged above pre-tax levels immediately after the repeal![1]

The example above shows two dangers.  Obviously, basing a policy on a faulty estimate of an important elasticity will get you in trouble.  But applying closed economy thinking to an increasingly open economy is an even greater and more insidious danger.

Let us consider one more example, briefly.  In the early 1980s, Canada imposed additional national taxes on cigarettes.  These taxes added a total of $3 (Canadian) to the cost per pack.  The government’s logic here was twofold. They hoped to raise revenues, since adult smokers have an inelastic demand for cigarettes. They also hoped to deter new smokers, particularly teens, with the higher prices. 

Sales plummeted.  Had the government incorrectly estimated the price elasticity of demand for cigarettes in Canada ?  Soon it became obvious that smoking was not falling nearly as much as the sales figures indicated.  Cross border sales in the US , particularly unregulated and untaxed sales on border Indian reservations, were booming.  Canada accused US tobacco companies of orchestrating (or at least facilitating) these sales, and sued for the lost tax revenues.  At the end of this section, we will do a quick, back of the envelop estimate of how much Canada should be compensated for this smuggling, assuming the courts found in Canada ’s favor.

What followed, to the present, is a classic case of inability to coordinate policies across national borders.  Canada appealed to the US to raise its taxes (which had declined from 42 percent of the price of a pack in 1951 to just 15 percent by the late 1980s), and/or police border smuggling.[2] The US refused.  After a decade of struggling with smuggling from the US, compounded by the involvement of indigenous peoples and, arguably, the US tobacco industry, Canada repealed most of the tax in 1994, bringing cigarette prices back into line with the US.  Some western provinces instituted their own taxes at that time, however, to avoid a consumer price fall.  Now state lawsuits in the US are being settled with provisions for sharp increases in taxes on cigarettes, with the proceeds to fund education efforts to prevent teen smoking.  Prices in the US rose by nearly a third in 1998, with US prices higher than those in the two most populous Canadian provinces of Ontario and Quebec .  Now Detroit shoppers are loading up on cigarettes in Windsor , Canada !  Canadian groups are calling for an increase in the cigarette tax in Canada , citing a loss of $1.98 billion in revenues by the provinces that rolled back the tax in 1994.[3]

Typically, non-economists estimate tax revenues lost by implicitly assuming a price elasticity of demand of zero (perfectly inelastic demand).  Thus, the $2 billion estimate above is probably based on a $2.50 per pack tax times current sales of 800 million packs per year. Even though this tax would roughly double the price of a pack of cigarettes, they assume no decrease in demand.  But even with an inelastic long run demand elasticity of 0.75, the actual loss in revenue is much lower, as shown below.  Note that this big proposed change in price means that we need to introduce the more sophisticated method of “arc elasticity,” described below. A rule of thumb is that you can use the normal method of calculating percentage changes in price and quantity, as long as the changes are less than 25%, and be confident that the error from this simplification is about 10% or less.  If the change you calculate is one-third or greater, you should use the more complex arc elasticity to avoid errors that are 20% or greater. 

The logic behind arc elasticity is simple.  Why should an increase from 1 to 2 be a 100% increase, while the decrease from 2 to 1 is just a 50% fall?  Suppose a firm sells 1000 units at $1, and raises the price to $2.  If the price elasticity of demand is (–)0.7, we would predict a fall in sales to 300 units.  Now suppose the firm cuts its price back to $1.  Naturally, sales will return to 1000 units at the new equilibrium.  But, using our simple methodology, we would predict an increase from 300 units to 300 * (50% * 0.7 +1) or just 405 units!  This amount of error is unacceptable.

With the arc elasticity, we divide the change in price or quantity by the average price or quantity to compute the percent change. 

[% change = (Pnew – Pold)/[(Pnew + Pold)/2].

Thus the increase in price from 1 to 2 and the decrease in price from 2 to 1 are both 67% changes [1/(3/2) = 2/3].  Sales would initially drop by 0.7 * 2/3 = 46.7%, then rise by the same amount back to 1000 when the price falls back to $1.  The trickiest part of using arc elasticities is not calculating the change in price, but in calculating the new quantity.  If we reduced 1000 to 533, then a 46.7% increase would clearly not raise it back to 1000. 

The formula for Qnew is Qold * (2 + %change)/(2–%change). 

Note first that if %change = 0, Qnew = Qold * 2/2 = Qold. That is good.  In our case, we have first a 46.7% decrease in Q.  Thus Qnew = 1000 * (2 + –0.467)/(2 – –0.467) = 1000 * 1.533/2.467 = 1000 * 0.621 = 621.  When the price falls, the new quantity will be Qnew = 621 * 2.467/1.533 = 621 * 1.61 = 1000 (actually 999.8, with rounding errors, but that is a whole lot better than the other method).  This is a more complex calculation, so we recommend that you:

Use it only when the increased accuracy is worth the effort (when you estimate a change in P or Q of greater than 1/3 by the simple method).

Check you work more carefully, especially checking first to make sure that you predict a fall in Q when P rises and vice versa!  

Packs sold without tax = $2 billion /$2.50 per pack = 800 million packs.  With the tax, the $2.50 increase in price to $5.00 is, in arc terms, an increase of (Pnew – Pold)/Paverage, or $2.50/$3.75 = 67 percent. 

Recall from before that we are assuming a long-run elasticity of 0.75.  This means that the price change will reduce demand by 0.75 = (% change in quantity demanded)/67%, thus % change in quantity demanded = 0.75 *67 = about 50 percent. 

Operationalizing the 50 percent decrease, in arc terms, means that

(Dnew –Dold)/Daverage = 50%.  If Dold = 800 million, we use our formula

Dnew = Dold * (2 –x)/(2+x), where x = % change in demand. 

Here, (2 – 0.5)/(2 + 0.5) = 1.5/2.5 = 0.6.  800 million packs times 0.6 equals 480 million.[4] If sales fall to 480 million, then tax revenues would be just 480 million times $2.50 = $1.2 billion.  If the tax increase again makes US cigarettes an attractive option for Canadian smokers, the public health benefits will lessen, and the tax revenues will decrease as well. Hence a “reasonable” compensation to request from the US tobacco companies, if it is found that they conspired to funnel cheap cigarettes to Canada , is around $1.2 billion per year less actual taxes collected in those years.

top