The Total-Revenue Test
The text suggests a test for demand elasticity based on the relationship between price and elasticity. As it turns out, there is a direct mathematical link between price, revenue, and elasticity.
What we require for our test is a relationship between the change in revenue and the change in price that brought it about. As hoped, this relationship will depend on the elasticity of demand.
Before we find this relationship, we require a slight modification of our elasticity formula. We begin by writing the demand relationship as Q = f(P). We know that the elasticity of demand is given by Ed = 
. (The minus sign accounts for the fact that demand elasticities are usually stated as positive values, even though 
is negative.) Taking the limit as D
P approaches 0, the elasticity becomes Ed = 
. Note that the first term, dQ/dP = f ’(P), is the (inverse of the) slope of the demand relationship. That is, elasticity of demand can be written as Ed = - f ’(P)
.
To continue, we note that revenue, R, is defined as price times quantity: R = PQ. If we substitute f(P) for Q, then R = Pf(P). The relationship we require is dR/dP. Using the multiplication rule for differentiation, dR/dP = f ’(P) x P + f(P). However, f(P) is just Q: dR/dP = f ’(P) x P + Q. By both multiplying and dividing by Q, this can be rewritten as dR/dP = Q(f ’(P)
+ 1). From our earlier result, we can now substitute the elasticity of demand for f ’(P)
to obtain dR/dP = Q(-Ed + 1) = Q(1 - Ed).
The total-revenue test is a straightforward application of this result: dR/dP = Q(1 - Ed). If demand is elastic, Ed > 1, then dR/dP < 0: price and total revenue move in opposite directions. If demand is inelastic, Ed < 1, then dR/dP > 0: price and total revenue move in the same direction. Finally, if demand is unit elastic, Ed = 1, then dR/dP = 0: an increase in price leaves total revenue unchanged.