Math 108: Notes on Elasticity of Demand
Percentage increases, Percentage rate of increase
A few months ago, Coca-cola in the second floor vending room of this building cost 75 cents per can.
Starting in September the price was $1 per can. The wrong way to look at this situation is “Twenty-five
cents is not much money, so the increase was small. ” The right way to look at the situation is “Dividing
the increase by the former price shows a 33% price increase and that’s big.” It’s the percentage increase,
not the actual increase, that should worry consumers.
Whenever you have a function y=f(x)relating two quantities, we know that f0(x)gives the rate of
change of ycompared to x. The
relative rate of change =f0(x)
f(x)and percentage rate of change =100 ∗f0(x)
f(x).
Note that these quantities probably depend on the xvalue you started with and should really be called the
relative and percentage rates of change starting at x.
Elasticity of Demand
Demand for gizmos is sensitive to unit price. An increase in price causes a drop in demand. Suppose
we have a demand equation x=f(p)where pis unit price and xis the number of gizmos that can be sold
at price p. The percentage rate of change in demand is
100 ∗f0(p)
f(p).
This predicts the percentage change in demand corresponding to a price increase of $1, and it is always
negative. Starting at a given price p, raising prices by $1 will result in a percentage increase in price of
100 ∗1
p. Economists define elasticity of demand starting at price level p to be the ratio
E=
percentage change in demand
percentage change in price
=
100 ∗f0(p)
f(p)
100 ∗1
p
=
p∗f0(p)
f(p)
.
Why use the absolute value? Because otherwise every entry in a table of demand elasticities would be
negative. Note that because our f0(p)<0, this is exactly the same as the book’s definition E=−p∗f0(p)
f(p).
Why Care About Elasticity of Demand?
Recall that our revenue (= income) for selling gizmos is given by
R=Revenue =unit price ∗number sold =p∗f(p)
where pis unit price and f(p)is the demand function. Our real interest is in the question “Will our revenue
rise if we increase prices slightly, starting at price level p?” In other words, is Ran increasing function
near p? In still other words, is R0(p)>0?
We use the product rule to find R0(p)and throw in some algebraic trickery to show how elasticity of
demand, E, is the key. At one point we will use the fact that p∗f0(p)
f(p)=−E. Here is the calculation:
R0=p∗f0(p) + 1∗f(p) = f(p)∗p∗f0(p)
f(p)+1=f(p)∗(−E+1).
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