Find the elasticity of demand (E) for the given demand function at the indicated values of p. Is the demand elastic, inelastic, or neither at the indicated values?
q=410 - 0.2 p^2
a.$21
b.$39
recall that
E = dq/dp * p/q
To find the elasticity of demand (E) for the given demand function, we need to use the elasticity formula:
E = (dq / dp) * (p / q)
Where:
dq / dp is the derivative of the demand function with respect to the price (p)
p is the price
q is the quantity demanded.
Let's start by finding the derivative of the demand function dq / dp:
dq / dp = -0.4p
Now, we can substitute the given values of p into the demand function and the derivative to find the quantity demanded (q) and the derivative at the indicated prices.
a. For p = $21:
q = 410 - 0.2 * (21^2)
q = 410 - 0.2 * 441
q = 410 - 88.2
q ≈ 321.8
dq / dp = -0.4 * 21
dq / dp = -8.4
E = (-8.4) * (21 / 321.8)
E ≈ -0.547
Based on the value of E, we can determine the elasticity of demand. If E > 1, the demand is elastic. If E < 1, the demand is inelastic. If E = 1, the demand is unit elastic. In this case, since E ≈ -0.547, the demand is inelastic.
b. For p = $39:
q = 410 - 0.2 * (39^2)
q = 410 - 0.2 * 1521
q = 410 - 304.2
q ≈ 105.8
dq / dp = -0.4 * 39
dq / dp = -15.6
E = (-15.6) * (39 / 105.8)
E ≈ -5.744
Again, since E ≈ -5.744, the demand is inelastic.
Therefore, in both cases, the demand is inelastic.