Answer the following questions for the price-demand equation. p + 0.005x = 30 A. Express the demand x as a function of the price p. B. Find the elasticity of demand, E(p). C. What is the elasticity of demand when p = 15 AND what does that mean? D. If the price is increased by 20%, what is the approximate change in demand?

Answer:

A. The price-demand equation given is p + 0.005x = 30. We need to express the demand x as a function of the price p. We can solve for x as follows:

p + 0.005x = 30 0.005x = 30 - p x = (30 - p) / 0.005 x = 6000 - 200p

Thus, the demand function, x(p), is x = 6000 - 200p.

B. To find the elasticity of demand E(p), we first need the derivative of x with respect to p. From our demand function, we get:

dx/dp = -200

The elasticity of demand is given by the formula:

E(p) = (dx/dp) * (p/x)

Let's substitute the values we have:

E(p) = (-200) * (p/(6000 - 200p))

C. To find the elasticity of demand when p = 15, we substitute p with 15 in the elasticity of demand formula we found:

E(p) = (-200) * (15/(6000 - 200*15))

We now simplify the expression:

E(15) = (-200) * (15/(6000 - 3000)) E(15) = (-200) * (15/3000) E(15) = (-200) * 0.005 E(15) = -1

The elasticity of demand when p = 15 is -1. This means that at this price level, the demand is unit elastic, which implies that a 1% change in price leads to an exactly 1% change in the quantity demanded.

D. If the price is increased by 20%, we want to know the approximate change in demand. Since the price elasticity of demand at p = 15 is -1, a 20% increase in price should result in approximately a 20% decrease in demand.

The exact change can be calculated by considering the new price and recalculating the demand using the demand function, but the approximate change in demand can be taken as a 20% decrease due to the unitary elasticity at p = 15.

Extra:

The concept of price elasticity of demand is a measure of the responsiveness of the quantity demanded of a good to a change in its price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. The elasticity can be categorized as follows:

- If E(p) < -1, the demand is elastic, meaning consumers are very responsive to price changes. - If E(p) = -1, the demand is unit elastic, implying an equal proportional change in quantity to that of price. - If -1 < E(p) < 0, the demand is inelastic, indicating consumers are less responsive to price changes.

Elasticity is important for businesses because it helps them understand how a change in their product's price could potentially affect their total revenue. In cases where demand is elastic, a decrease in price may lead to a higher total revenue, while for inelastic demand, a price increase might not significantly decrease sales, thus potentially increasing total revenue.