Question

A fried chicken franchise finds that the demand equation for its new roast chicken product, "Roasted Rooster," is given by p = 45 / q 1.5

where p is the price (in dollars) per quarter-chicken serving and q is the number of quarter-chicken servings that can be sold per hour at this price.

1) Express q as a function of p.

2) Find the price elasticity of demand when the price is set at $4.00 per serving.

Answer 1

1.
`q=((45)/(p))^{(2)/(3)}`

2.
`E_d=-(2)/(3)`

Explanation:

The given demand equation is

`p=(45)/(q^(1.5))`

where p is the price (in dollars) per quarter-chicken serving and q is the number of quarter-chicken servings that can be sold per hour at this price.

Part 1 :

We need to Express q as a function of p.

The given equation can be rewritten as

`q^(1.5)=(45)/(p)`

Using the properties of exponent, we get

`q=((45)/(p))^{(1)/(1.5)}`
`[\because x^n=a\Rightarrow x=a^{(1)/(n)}]`

`q=((45)/(p))^{(2)/(3)}`

Therefore, the required equation is
`q=((45)/(p))^{(2)/(3)}`.

Part 2 :

`q=(45)^{(2)/(3)}p^{-(2)/(3)}`

Differentiate q with respect to p.

`(dq)/(dp)=(45)^{(2)/(3)}(-(2)/(3))(p^{-(2)/(3)-1}})`

`(dq)/(dp)=(45)^{(2)/(3)}(-(2)/(3))(p^{-(5)/(3)})`

`(dq)/(dp)=(45)^{(2)/(3)}(-(2)/(3))(\frac{1}{p^{(5)/(3)}})`

Formula for price elasticity of demand is

`E_d=(dq)/(dp)* (p)/(q)`

`E_d=(45)^{(2)/(3)}(-(2)/(3))(\frac{1}{p^{(5)/(3)}})* \frac{p}{(45)^{(2)/(3)}p^{-(2)/(3)}}`

Cancel out common factors.

`E_d=(-(2)/(3))(\frac{1}{p^{(5)/(3)}})* \frac{p}{p^{-(2)/(3)}}`

Using the properties of exponents we get

`E_d=-(2)/(3)(p^{-(5)/(3)+1-(-(2)/(3))})`

`E_d=-(2)/(3)(p^(0))`

`E_d=-(2)/(3)`

Therefore, the price elasticity of demand is -2/3.