Answer:
a) -0.842
b) $0
c) 11,664 cases
Step-by-step explanation:
a)
The price elasticity of demand E at a given point [tex]\large (p_1,q_1)[/tex] is defined as
[tex]\large E=\frac{p_1}{q_1}.\frac{\text{d}q}{\text{d}p}(p_1)[/tex]
and in this case, it would measure the possible response of tissues demand due to small changes in its price when the price is at [tex]\large p_1[/tex]
When the price is set at $32 the demand is
[tex]\large q=(108-32)^2=5,776 [/tex]
cases of tissues, so
[tex]\large (p_1,q_1)=(32,5776)\Rightarrow \frac{p_1}{q_1}=\frac{32}{5776}=0.00554[/tex]
Also, we have
[tex]\large \frac{\text{d}q}{\text{d}p}=-2(108-p)\Rightarrow \frac{\text{d}q}{\text{d}p}(32)=-2(108-32)=-152[/tex]
hence
[tex]\large E=\frac{p_1}{q_1}.\frac{\text{d}q}{\text{d}p}(p_1)=0.00554(-152)=-0.842[/tex]
That would mean the demand is going down about 0.842% per 1% increase in price at that price level.
b)
When the price is $108 the demand is 0, so the price should always be less than $108.
On the other hand, the parabola
[tex]\large q=(108-32)^2=5,776 [/tex] is strictly decreasing between 0 and 108, that means the maximum demand would be when the price is 0.
c)
When the price is 0 the demand is
[tex]\large (108)^2=11,664[/tex] cases
The elasticity for p = 32 is E = -0.84.
The maximum revenue is obtained when p = 32, and the demand for that price is q = 5,184.
It is given by:
[tex]E = \frac{p}{q} *\frac{dq}{dp} (p)[/tex]
if p = $32, then:
q(32) = (108 − 32)^2 = 5,776
and:
dq/dp = -2*(108 - p)
Evaluating that in p = 32 we get:
-2*(108 - 32) = -152
Then the elasticity is:
E = (32/5,776)*-152 = -0.84.
The revenue is equal to the demand times the price, so we get:
R = p*q = p*(108 - p)^2 = p*(p^2 - 216p + 11,664)
To maximize this we need to find the zeros of the derivation, we have:
R' = 3p^2 - 2*216*p + 11,664
The zeros of that equation are given by:
[tex]p = \frac{432 \pm \sqrt{(-432)^2 - 4*3*11,664} }{2*3} \\\\p = (432 \pm 216)/6[/tex]
Notice that we need to use the smaller of these values, so the demand never becomes zero, then we use:
p = (432 - 216)/6 = 36
(the other root gives p = 108, so that is a minimum).
c) For this price, the number of cases demanded is:
q = (108 - 36)^2 = 5,184
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