1. In-season: The family should keep selling 17 car washes per day. The marginal revenue of selling one more car wash is greater than the marginal cost.
2. Off-season: The family should shut down the car wash. The marginal revenue of selling one more car wash is less than the marginal cost.
To maximize profit, the family should follow the golden rule of profit maximization: produce until the marginal revenue equals the marginal cost. This means that they should continue to produce as long as the extra revenue they generate from selling one more car wash is greater than the extra cost of producing that car wash.
The fixed cost of the car wash is $15 per day. This is the cost of the loan that the family took out to start the business. The variable cost of the car wash is the cost of labor and supplies. This cost increases as the number of car washes sold increases.
When the beach resort is in-season, the family can charge a price of $14 per car wash. When the resort is off-season, the family can only charge a price of $4.67 per car wash.
To answer the family's question, we need to calculate the marginal revenue and marginal cost of selling car washes.
Marginal revenue is the extra revenue generated from selling one more car wash. It is equal to the price per car wash.
Marginal cost is the extra cost incurred from producing one more car wash. It is equal to the variable cost per car wash.
In-season:
Marginal revenue = $14 per car wash
Marginal cost = $10.67 per car wash
Off-season:
Marginal revenue = $4.67 per car wash
Marginal cost = $10.67 per car wash
In-season: Keep selling 17 car washes per day.
Off-season: Shut down the car wash.