VI.  ELASTICITY AND THE MEASUREMENT OF SUPPLY AND DEMAND


A. Responsiveness of One Variable to Another

In economics, as with the physical and biological sciences, we are interested in measuring how much of an effect one factor has on another, that is measuring how responsive one variable is to changes in another variable.  For instance, we are interested with measuring the size of a change in quantity demanded or quantity supplied from a give change in price.  We are also interested in measuring the size of a change in quantity demanded brought about by a given change in such factors as income, prices of substitutes, and prices of complements.

There is a concept which you should be familiar with from algebra­­the concept of slope.  As you should recall, slope measures rise over run, or how much something goes up or down as a related factor increases by one unit.  A slope shows the change in some factor we might call "y," brought about by a one unit change in some other factor we might call "x."  The slope is denoted , where "d" means "change in ______," or "d ifference in ______."   You will recall from algebra that a linear equation is of the form

11)  y = mx + b,

where "b" is the constant, or the y­axis intercept, or the value that y would obtain if x were zero; and "m" is the slope of the equation, m is also equal to dy/dx . 

B. Elasticities

We could use slope as a measure of the effect that one variable has on another, but we encounter a problem, we find that we cannot compare the change in the consumption of gasoline from a one dollar change in price with the change in consumption of automobiles form a one dollar increase in price.  The reason is that the units these are measured in are not comparable, the problem of adding apples and oranges.  The change in consumption of gasoline divided by the change in price of gasoline is measured in units of gallons per dollar (or gal./$).  The change in consumption of automobiles divided by a one dollar change in price is measured in automobiles per dollar (or cars/$).  These two measures are in different units and cannot be compared in a meaningful way.  Likewise the change in consumption of gasoline from a one dollar change in price in the U.S. cannot be compared with the change in consumption of gasoline from a one pound change in price in the U.K.  Again, it is a matter of incomparable units. 

This can be overcome by washing the units out of the measure.  This is done by dividing the change in quantity demanded by the initial value of quantity demanded and by dividing the change in price by the initial value of price.  The units drop out when change in quantity is divided by quantity, or when, for instance, change in gallons of gasoline is divided by gallons of gasoline.  The number becomes a percent, a percentage change, a "unitless" number.  Dividing one "unitless" number by another again produces a "unitless" number.  

Elasticity is the percentage change in the dependent variable from a one percent change in some independent variable, holding other variables constant.  The price elasticity of demand can be seen as

12) E_d = %d QD/ % d P = (d QD/QD_avg)/(d P/P_avg),

where QD_avg is the average quantity demanded and P_avg is the average price of the good.  We can look at the change in quantity demanded from one point to another and the elasticity, E_d , is

Notice that the 2's for the averages cancel out in the second step.

Do not get the idea that the elasticity and slope are the same, or even reciprocals of one another.  The slope of a demand curve is 

14) slope of demand = (P₁ - ­ P₂)/(QD₁ -­ QD₂),

while the reciprocal of the slope is

15) reciprocal of slope of demand = (QD₁ -­ QD₂)/(P₁ - ­ P₂),

and elasticity is

13) E_d = [(QD₁ -­ QD₂)/(P₁ - ­ P₂)] x [(P₁ + P₂)/(QD₁ + QD₂)].

Notice that the reciprocal of the slope is the first part of the elasticity  formula before the multiplication sign, but that the two are not equal, except when the sum of the two prices is equal to the sum of the two quantities.

Next we should note that economists often take the absolute value of the price elasticity of demand to make discussion easier. In this way, economists are able to simplify discussions of the elasticity number by getting rid of the naturally negative sign. Probably the best way to tell you why economists change the sign of the number is to share an e-mail message from a student in a previous class and my reply to that student:

Dear Dr. Coats,

I am in your 5T class, and today towards the end of the period you
discussed elasticity. I read the chapter, but I don't quite understand the
formula you gave us. Why take the absolute value of change in price and
change in quantity? Why can't you have a negative result? Wouldn't the
negative result mean that demand is low?
I hope you can answer my questions.

Thank you,

Student (the student signed her name)

Dear Student:

The main reason for taking the absolute value of elasticity is for
simplicity of discussion. I used to discuss elasticity without taking
the absolute value, but here's the problem:

The price elasticity of demand is virtually always negative, for our
purposes, exceptions are so rare that we can, at our introductory level
of discussions, ignore the cases where the elasticity of demand is not
really negative. The more negative the elasticity becomes, the more
elastic demand becomes. This last concept is difficult/confusing for
some. As a result, it becomes easier for us to discuss elasticity by
having an "understood" negative sign in front of the elasticity. You may
wish to think of it as just converting a number that is, for our
purposes, always negative, to a positive number by placing a negative
sign in front of the formula. It just makes discussions easier.

One cannot tell how high or low demand is by looking at the
price elasticity of demand. Price elasticity tells us how sensitive
buyers are to price changes. It really makes no sense to talk of demand
as being high or low, except in comparison to some other demand. That
is, we can only say that demand is higher or lower than it was at some
previous point in time. So it is rather meaningless to talk of the
demand for motorcycles being higher or lower than the demand for trucks.
The reason is that the quantities demanded of motorcycles and trucks are
measured in different units, and so incomparable. Even talking about the
demand for gasoline and water are incomparable, the quantity demanded of
water is measured in gallons of water, while the quantity demanded of
gasoline is measured in gallons of gasoline--two different and
incomparable items.

I hope that this helps. If not, please write back.

Morris Coats

C. Elasticity Ranges

There are five ranges of price elasticity of demand that we should note here, where:

a)  perfectly elastic demand    Ed = infinity,

b)  perfectly inelastic demand  Ed = 0,

c)  relatively elastic demand   Ed > 1,

d)  relatively inelastic demand Ed < 1, and

e)  unitary elastic demand      Ed = 1. 

The elasticity of demand is determined by how many alternatives and how good those alternatives are that one has for a given product, because the more good alternatives one has for a particular good, the easier one can alter one's behavior in the face of changed circumstances.  Also, the longer time period under consideration, the more elastic demand will be because one has more opportunities to alter behavior. Another factor in determining the elasticity of demand for an item is the proportion of one's annual budget that one spends on that item.

D. Other Elasticities of Interest

For certain problems, we may be interested in other elasticities, the responsiveness of other pairs of variables.  Later we will discuss the price elasticity of demand and the price elasticity of supply in looking at the effects of excise taxes.  We are also concerned with the responsiveness of quantity demanded to other variables, such as income and prices of other related goods. 

The price elasticity of supply shows how responsive quantity supplied is to changes in price.  Like the price elasticity of demand, the more alternatives a producer has and the longer time period under consideration, the higher will be the price elasticity of supply.  The formula is much like the formula for the price elasticity of demand as in 19), with the only difference being that QD is replaced by QS (or quantity supplied):

16)  E_s = [(QS₁ -­ QS₂)/(P₁ - ­ P₂)] x [(P₁ + P₂)/(QS₁ + QS₂)].

The income elasticity of demand shows how responsive quantity demanded is to changes in income.  A positive income elasticity indicates that a good is a normal good, while a negative income elasticity indicates that a good is an inferior good.  It is also useful in determining the effect on the poor of a tax on a particular items, such as cigarettes, beer, cars, and jewelry.  The formula for the income elasticity of demand is like the one for price elasticity, except that the P's (for price) are replaced with Y's (for income):

17)  E_Y = [(QD₁ - ­ QD₂)/(Y₁ - ­ Y₂)] x [(Y₁ + Y₂)/(QD₁ + QD₂)].

The cross elasticity of demand shows how responsive the quantity demanded of good "a" is to changes in the price of some related good, "b."  Positive cross elasticities indicate that the two goods are substitute goods, while negative cross elasticities indicate that the two goods are complements.  The formula for the cross elasticity of demand is given by 

18) E_d^ab = [(QD₁^a - ­ QD₂^a)/(P₁^b - ­ P₂^b)] x [(P₁^b + P₂^b)/(QD₁^a + QD₂^a)].

E. Consumer Expenditures, Seller Revenues and Price Elasticity of Demand

The price elasticity of demand also can tell us what will happen to the firm's revenues as the price of the good is changed.  A firm's revenues from a particular product will be the number of items sold times the selling price of that item, for example, if you sell 100 units of a good at $10 per unit, you will receive $1000.  This can be stated in the following equation

19) R = P x QD,

where R is revenues and P and QD are price and quantity demanded, respectively.  Since price and quantity demanded are inversely related, an increase in product price will not always increase revenues.  I recall two young girls in my hometown who tried to make some money selling pears from their pear tree.  They set up their stand with a price of $5 per pear on their sign.  Needless to say, their revenues were $0.  When they dropped their price to $0.10 per pear, their revenues went up.  Tuition could get so high here that any further increase could cause so many students to go elsewhere or not go to college at all that the school could actually reduce the money it gets from students by increasing tuition.

If the percentage increase in price is greater than the percentage decrease in quantity demanded, a price increase will cause a revenue decrease.  But if the percentage drop in quantity demanded exceeds the percentage increase in price, a price increase will result in a drop in revenues.  Remember from 12) above

12)  E_d = %d QD/ %d P,

so that if %d P > ­%d QD, E_d will be between 0 and 1, an inelastic demand.  If %d P < ­%d QD, E_d > 1, and so demand is said to be elastic.  So we see that if demand is elastic, a price increase will lead to a revenue decrease and a price decrease will lead to a revenue increase. If demand is inelastic, price and revenue move in the same direction.  If demand is unitary elastic, price changes leave revenue unaltered.  This is summarized in Table 3 .

TABLE 3

                                                              
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