Slim city is a town resided by slim people, with no health club in town. LA Fatness has estimated that the annual demand for their Fatness Club in this town to be Q = 160 - P. Every Club member has a marginal cost of $40, and the annual fixed cost of the business is $500 (assume a linear cost function).
LA Fatness starts the first Fatness Club in town. What is the optimum price & membership quantity?
To find the optimum price and membership quantity for LA Fatness Club in Slim city, we need to determine the profit-maximizing level of output where marginal revenue equals marginal cost.
First, let's write the profit function for LA Fatness Club. Profit (π) can be calculated as:
π = Total Revenue - Total Cost
Total Revenue (TR) is the product of price (P) and quantity sold (Q), and the demand function given is Q = 160 - P. Therefore:
TR = P * Q = P * (160 - P)
Total Cost (TC) can be divided into two components: fixed cost (FC) and variable cost (VC). The given fixed cost is $500, and the variable cost is given as $40 per club member. Since the demand function represents the quantity of club members, the variable cost can be written as:
VC = 40 * Q = 40 * (160 - P)
Thus, the total cost function is:
TC = FC + VC = 500 + 40 * (160 - P)
Now, we can express the profit function as:
π = TR - TC = P * (160 - P) - (500 + 40 * (160 - P))
To find the optimum price and membership quantity, we need to differentiate the profit function with respect to P and set it equal to zero to solve for P. Let's calculate:
dπ/dP = 0
d(P * (160 - P) - (500 + 40 * (160 - P)))/dP = 0
160 - 2P + 40 = 0
-2P + 200 = 0
-2P = -200
P = 100
So the optimum price for the Fatness Club in Slim city is $100.
To find the membership quantity, we can substitute the optimum price back into the demand function:
Q = 160 - P = 160 - 100 = 60
Therefore, the optimum membership quantity for the Fatness Club is 60.
In summary, the optimum price for the Fatness Club in Slim city is $100, and the optimum membership quantity is 60.