Question

Exercise 7. Elasticity and Revenue (Constant Elasticity Demand)

The market demand function is X(p) = 100/p².

1. What is the elasticity, ε(p)?

2. If p=1, what is market demand, X? What is sales revenue R=p∙X?

3. If price increases to p’=2, what is X’? What is R’?

4. The natural logarithm of the market demand is ln(X) = ln(100) -2ln(p). Take the derivative of ln(X) with respect to ln(p). What is ∂ln(X)/∂ln(p)? How does it compare to the elasticity result you found in part a?

Answer 1

a. ε(p) = (dX/dp)*(p/X)
X(p) = 100/p2 = 100p-2
dX/dp = -2(100p-2-1) = -200p-3
So, ε(p) = (-200p-3)(p/100p-2) = -2p-2/p-2 = -2
So, ε(p) = -2.

b. X(1) = 100/12 = 100
R = p*X = 1*100 = 100

c. X'(2) = 100/22 = 100/4 = 25
R' = p'*X' = 2*25 = 50

d. ln(X) = ln(100) -2ln(p)
∂ln(X)/∂ln(p) = -2
So, ∂ln(X)/∂ln(p) equals the value of elasticity.