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

qS=10p−200

1. Find the inverse supply function.

We need to solve this equation for p:

qs=10p−200Original supply equationqs+200=10pAdding 200 to both sides110qs+20=pDividing both sides by 10

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

The formula for price elasticity of supply is ϵ=1slope∗pq. 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:

qs=10p−200qs=10(25)−200qs=250−200qs=50

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

ϵ=1slope∗pqϵ=1(110)∗(25)(50)ϵ=10∗0.5ϵ=5

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 ϵ=1slope∗pq. 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:

qs=10p−200qs=10(50)−200qs=500−200qs=300

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

ϵ=1slope∗pqϵ=1(110)∗(50)(300)ϵ=10∗16ϵ=106≈1.67

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%.