A bakery works out a demand function for its chocolate chip cookies and finds it to be q = D(x)= 760-13x​, where q is the quantity of cookies sold when the price per​ cookie, in​ cents, is x. Required: a. Find the elasticity. b. At what price is the elasticity of demand equal to 1? c. At what prices is the elasticity of demand elastic? d. At what prices is the elasticity of demand inelastic?

A) Find the elasticity.

Elasticity of demand is a measure of how responsive the quantity demanded is to a change in price. It can be represented as the percentage change in quantity demanded divided by the percentage change in price. The formula for elasticity (E) at a particular point (x, q) on the demand curve is given by:

E = (dq/dx) * (x/q)

where dq/dx is the derivative of the demand function with respect to the price.

Given the demand function q = 760 - 13x, we can differentiate it with respect to x to obtain dq/dx:

dq/dx = -13

Now we can substitute dq/dx and the corresponding price (x) and quantity (q) into the elasticity formula:

E = (-13) * (x / (760 - 13x))

B) At what price is the elasticity of demand equal to 1?

To find the price where the elasticity of demand is equal to 1, set the elasticity formula equal to 1 and solve for x:

1 = (-13) * (x / (760 - 13x))

To solve for x, we will cross multiply:

760 - 13x = -13x

From here, it seems there is no meaningful solution because the x terms cancel each other out on both sides, which would indicate that the elasticity of demand cannot exactly equal 1 at any given price using this linear demand function. However, there might have been a mistake in the initial reasoning. Typically, for a linear demand curve, the elasticity of demand is equal to 1 (unit elastic) at the midpoint of the demand curve. Let's check if there's a calculation error.

We should instead solve for x like this:

760 - 13x = 13x

Combining the x terms:

760 = 26x

Now, divide by 26 to get x:

x = 760 / 26 x ≈ 29.23

At a price of approximately 29.23 cents per cookie, the elasticity of demand is equal to 1.

C) At what prices is the elasticity of demand elastic?

The demand is elastic when the elasticity is greater than 1, which means that consumers are very responsive to price changes.

Since the elasticity for the linear demand function q = 760 - 13x is:

E = (-13) * (x / (760 - 13x))

Demand is elastic (E > 1) when:

-13 * (x / (760 - 13x)) > 1

Using the fact that in a linear demand curve, demand is elastic below the midpoint (where E = 1), we can say that for any price lower than 29.23 cents (the calculated midpoint), the demand will be elastic.

D) At what prices is the elasticity of demand inelastic?

The demand is inelastic when the elasticity is less than 1, which means that consumers are not very responsive to price changes.

Using the mid-point concept again, inelastic demand (E < 1) occurs above the midpoint:

For any price higher than 29.23 cents per cookie, the demand will be inelastic.