Science:Math Exam Resources/Courses/MATH104/December 2015/Question 05 (c)

MATH104 December 2015

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Question 05 (c)
A tailor is currently producing 80 suits per month and sells them for $100 per suit. His monthly demand curve is given by $q=100-2{\sqrt {p}}$. Find the current price elasticity of demand and use it to decide whether price should be raised or lowered to increase his revenue.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Recall that the price elasticity of demand is given by the formula $\epsilon ={\frac {p}{q}}{\frac {dq}{dp}}$ where $p$ is the price of the good and $q$ is the demand (as a function of $p$).

Hint 2
Recall that... ...if $|\epsilon |>1,$ the good is price elastic, so the price should be decreased ...if $|\epsilon |<1,$ the good is price inelastic, so the price should be increased ...if $|\epsilon |=1,$ the good is price unit elastic, so the price should not be changed (since the current price maximizes revenue)

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We calculate the price elasticity of demand according to the formula $\epsilon ={\frac {p}{q}}{\frac {dq}{dp}}$ given $q(p)={\color {OrangeRed}100-2{\sqrt {p}}}$ and $p={\color {OliveGreen}100}:$ ${\begin{aligned}\epsilon &={\frac {p}{q}}{\frac {dq}{dp}}\\&={\frac {p}{\color {OrangeRed}100-2{\sqrt {p}}}}\cdot {\frac {d}{dp}}({\color {OrangeRed}100-2{\sqrt {p}}})\\&={\frac {p}{100-2{\sqrt {p}}}}\cdot -2\cdot {\frac {1}{2{\sqrt {p}}}}\\&={\frac {-p}{100{\sqrt {p}}-2p}}\\&={\frac {-{\color {OliveGreen}100}}{100{\sqrt {\color {OliveGreen}100}}-2\cdot {\color {OliveGreen}100}}}=-{\frac {100}{800}}={\color {blue}-{\frac {1}{8}}}.\end{aligned}}$ Since $|\epsilon |<1$, the tailor should increase the price of their suits.