Elasticity Reading

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Next to comparative advantage, elasticity is probably one of the most difficult concepts to comprehend in economics.  There are a number of teaching methods that have been tried to help make elasticity more understandable.  I've seen many of these methods and found that most of them are difficult for students to understand.

When I first taught Econ 105 in the fall semester 1999, I introduced two different approaches on how to find elasticity...and thoroughly confused the entire class.  I have decided to create my own method (similar to the method of comparative advantage that you learned).  This reading is intended to provide an adequate introduction to elasticity, show you what to read and what NOT to read in Mankiw's text, and hopefully give you enough of a foundation to allow you to understand the basics for the next class lecture.

First, to avoid confusion, do NOT read page 92 of our text...this is the part that seemed to confuse everyone.

Now...on to the meat of the reading:

Many people questioned if the slope of the curves was important.  Indirectly, it is, and we will see why in a moment.  By definition, elasticity is the measure of the responsiveness of quantity demanded or quantity supplied to one of its determinants (determinants were discussed in the last section e.g. income, market price, related goods, etc.).  By "responsiveness" we mean "how much quantity demanded or quantity supplied changes with respect to a change in something else."  For example, suppose the price of a good increases 10%.  If we were examining price elasticity of demand, we would see how much quantity demanded decreases with respect to that corresponding 10% increase in price (remember that we expect quantity demanded to decrease as price increases).  The number that is associated with elasticity is actually the ratio of the percentage changes.

In everyday terms, price elasticity of demand would be:
 

           % change in quantity demanded

                    % change in price
If the price increased 10% and the quantity demanded decreased by 5%, then the elasticity would be 0.5  [5/10].  If quantity demanded changed by 20%, then the elasticity would be 2 [20/10].  Note that we take the absolute value of the above...there are no negative elasticities.

Suppose you are told that the price elasticity for demand at a certain point is 0.5 .  What does this mean?  If price were to rise 10%, quantity demanded would fall by 5%.  Be careful...we could also say that, at that point, a 1% increase in price will decrease quantity demanded by 0.5% (or a 200% increase in price will decrease quantity demanded by 100%).  Since we do not take the sign into account, we could also say that a 1% decrease in price will increase quantity demanded by 0.5%.  The point to remember is that quantity demanded will change only half as much as price will.

What if we had a price elasticity of demand of 2?  Quantity demanded will change twice as fast as price changes.  How about an elasticity of .923456?  Quantity demanded will change .923456 as fast as price changes.

Whenever price elasticity of demand is greater than 1, we say that demand is elastic at that point.  Quantity demanded is very responsive to changes in price.  When it is less than 1, we say demand is inelastic...the quantity demanded is not very responsive to changes in price.

What about the case where elasticity = 1?  This is a special case, called unit elastic.  At that point, quantity demanded changes at the the same proportion as price.  If we find a point on the demand curve that is unit elastic, it has some interesting characteristics that we will discuss later.

The next item to discuss is actually calculating elasticity.  It involves some fairly simple math to set up.  The formula for figuring out the % change in something is as follows:
 

    (X2 - X1)

        X1
For example, if the price increased from $1.00 to 1.25, we would have:
 
      (1.25 - 1.00)

           1.00
or 25%.

Now, let's put the definition of price elasticity of demand in mathematical terms:
 

  ( Qd2 - Qd1)

        Qd1
    ( P2 - P1)

        P1
If we were to perform a slight modification to the above (invert the denominator and multiply), we get:
 
( Qd2 - Qd1)   *     P1
 ( P2 - P1)      *      Q1
Look at what we have if we take the above in two parts:
Part 1.
( Qd2 - Qd1)

 ( P2 - P1)
and
Part 2.
P1

Q1


For those of you who have taken calculus, price elasticity of demand is dQ/dP * P/Q .

Remember that, when we graph, Price (P) is always on the Y axis, and Quantity (Q) is always on the X axis.  Part 1 is actually the inverse of the slope of the demand curve.  If we assume that the demand curve is a line, then Part 1 is a constant number (if the demand curve has the equation Y = mX + B (or P = mQ + B...same thing), Part 1 is 1/m ).  For this Principles class, we will only examine linear supply and demand curves, so Part 1 will always be a constant.  Like opportunity costs, we will assume that this number will be positive, regardless of the sign.

Part 2  (P1 / Q1) is probably where the nature of most of the confusion lies.  There is one price and one quantity in this second part...it refers to a point on the curve.  Part 2 will change as you move along the demand curve.  If you start at a low price and high quantity (remember, the law of demand states that price and quantity are inversely related),  then P/Q is a small number.  As we move up the demand curve, P/Q gets larger (as we go up the demand curve, P increases and Q decreases).

Now for a quick algebra review...if we multiply a positive, constant number by a variable, then the answer depends on that variable...but we can safely say that the answer will increase if the value of that variable increases.  For example, consider the term 2x.  If x=5, then the answer would be 10.  If x were increased to 6, the answer would be 12.  It also works the opposite way -- if we decrease the variable, then the answer will decrease.  If x=3, the answer goes down to 6.  Pretty simple, right?

The exact same holds true for elasticity.  If we combine Parts 1 and 2, we have the exact same thing.  Part 1 is constant (the inverse of the slope), and Part 2 is a variable.  We can safely assume though, that the answer will increase if the value of the variable increases.  How does this help us?  It was mentioned earlier that P/Q increases as we move up the demand curve.  We can now say that the elasticity increases as we move up the demand curve.