The supply of bicycle rentals in a small town is given by:

\[q_S=10p-200\]

1. Find the inverse supply function.

We need to solve this equation for \(p\):

\[\begin{align*} q_s &= 10p - 200 && \text{Original supply equation}\\ q_s + 200 &= 10p && \text{Adding 200 to both sides}\\ \frac{1}{10}q_s + 20 &= p && \text{Dividing both sides by 10}\\ \end{align*}\]

2. What is the price elasticity of supply at a price of $25?

The formula for price elasticity of supply is \(\epsilon = \frac{1}{slope} * \frac{p}{q}\). We know the slope and \(p\), so we first have to find the quantity supplied, \(q\) at $25.

We can find this by plugging in \(p=25\) into the original supply function:

\[\begin{align*} q_s & = 10p-200 \\ q_s & = 10(25) - 200 \\ q_s & = 250 - 200\\ q_s &= 50\\ \\ \end{align*}\]

Now we can take everything and plug it into the elasticity formula:

\[\begin{align*} \epsilon & = \frac{1}{slope} * \frac{p}{q} \\ \epsilon & = \cfrac{1}{\left(\frac{1}{10}\right)} * \frac{(25)}{(50)} \\ \epsilon & = 10 * 0.5 \\ \epsilon & = 5\\ \\ \end{align*}\]

This is relatively elastic: for every 1% increase (decrease) in price, quantity supplied will increase (decrease) by 5%.

3. What is the price elasticity of supply at a price of $50?

The formula for price elasticity of supply is \(\epsilon = \frac{1}{slope} * \frac{p}{q}\). We know the slope and \(p\), so we first have to find the quantity supplied, \(q\) at $50.

We can find this by plugging in \(p=25\) into the original supply function:

\[\begin{align*} q_s & = 10p-200 \\ q_s & = 10(50) - 200 \\ q_s & = 500 - 200\\ q_s &= 300\\ \\ \end{align*}\]

Now we can take everything and plug it into the elasticity formula:

\[\begin{align*} \epsilon & = \frac{1}{slope} * \frac{p}{q} \\ \epsilon & = \cfrac{1}{\left(\frac{1}{10}\right)} * \frac{(50)}{(300)} \\ \epsilon & = 10 * \frac{1}{6} \\ \epsilon & = \frac{10}{6} \approx 1.67\\ \\ \end{align*}\]

This is also relatively elastic (but less elastic than at $25): for every 1% increase (decrease) in price, quantity supplied will increase (decrease) by 1.67%.