Question

A local gym plans to use third-degree price discrimination to set its family and corporate rates. The demand for family memberships is Q=984-20P and the demand for corporate memberships is Q=2070-50P. The gym’s cost is TC=12,000+10Q. 1. How many memberships of each will the firm sale under third-degree price discrimination? 2. What price will the firm charge for each type of memberships under third-degree price discrimination? 3. What profits will the firm make in total under third-degree price discrimination? 4. If the law goes not allow the gym to price discriminate, what profits will the firm make in total under uniform pricing (hint: add the original demands to find the total demand)?

Answer 1

The gym employs third-degree price discrimination to set distinct prices for family and corporate memberships, calculating the profit-maximizing price and quantity for each group by setting marginal revenue equal to marginal cost and using the demand curves. Total profits are determined by subtracting total costs from total revenues. If unable to price discriminate, the gym uses uniform pricing based on combined demand.

Step-by-step explanation:

The local gym's decision to use third-degree price discrimination allows it to set different prices for different segments, specifically family and corporate memberships, based on their respective demand curves. To find the quantity sold and the price for each segment, the gym needs to maximize profit for each segment separately:

1. For families, the demand is Q = 984 - 20P.
2. For corporates, the demand is Q = 2070 - 50P.
3. The total cost for the gym is TC = 12,000 + 10Q.

To maximize profits, the gym must calculate the marginal cost (MC) and marginal revenue (MR) for each segment and set MR equal to MC. The gym then uses the demand curve to find the profit-maximizing price.

If the gym cannot price discriminate due to legal restrictions, it will apply uniform pricing. In this case, the total demand is the sum of the individual demands. The gym then identifies the profit-maximizing quantity and price for the combined demand.

For both scenarios, profits are calculated by subtracting total costs from total revenue, where total revenue is the price times the quantity sold for each segment.

Answer 2

The firm will sell 1,264 family memberships and 1,920 corporate memberships.

The firm will charge -0.5 for family memberships and 0.2 for corporate memberships.

The total profit that the firm will make is -90.

How to solve

1. Determining the Number of Memberships

To maximize profit, the firm should set the prices for each type of membership such that the marginal revenue from each type is equal to the marginal cost. The marginal revenue for each type of membership can be found by taking the derivative of the demand function with respect to price:

`MR_(f) = (dQ_(f))/(dP_(f)) = -20M\\R_(c) = (dQ_(c))/(dP_(c)) = -50`

The marginal cost is simply the cost of producing one additional unit of output, which is 10 in this case. Therefore, we have the following system of equations:

`MR_(f) = MC = 10\\MR_(c) = MC = 10`

Solving for the prices, we get:

`P_(f) = (MR_(f))/(-20) = -0.5\\P_(c) = (MR_(c))/(-50) = 0.2`

Now, we can plug these prices back into the demand functions to find the quantities:

`Q_(f) = 984 - 20P_(f) = 1264\\Q_(c) = 2070 - 50P_(c) = 1920`

Therefore, the firm will sell 1,264 family memberships and 1,920 corporate memberships.

2. Calculating the Prices

The prices for each type of membership have already been calculated in step 1:

`P_(f) = -0.5\\P_(c) = 0.2`

3. Determining the Profit

The profit for each type of membership can be found by subtracting the marginal cost from the marginal revenue:

`\pi_(f) = MR_(f) - MC = -20 - 10 = -30\\\pi_(c) = MR_(c) - MC = -50 - 10 = -60`

The total profit is the sum of the profits from each type of membership:

`\pi = \pi_(f) + \pi_(c) = -30 - 60 = -90`

Therefore, the total profit that the firm will make under third-degree price discrimination is -90.