Slick Shades has a constant marginal cost of production equal to $80 and the distributors have a constant marginal cost of distribution equal to $30. If Slick Shades is producing the profit- maximizing number of sunglasses (in hundreds)and charging the profit- maximizing wholesale price, what is the retail price?
A)$160
B)$180
C)$170
D)$200

Question

Quizplus · Accepted Answer

To find the profit-maximizing wholesale price, we need to find the price that equates the marginal revenue with the marginal cost. Let's assume that Slick Shades sells x hundred sunglasses at the wholesale price of y dollars per unit. Then the total revenue for Slick Shades is given by:

TR = xy

The marginal revenue is given by the derivative of the total revenue with respect to x:

MR = dTR/dx = y

The marginal cost of production is $80 per hundred sunglasses, so the total cost of production for x hundred sunglasses is:

TC = 80x

The marginal cost of distribution is $30 per hundred sunglasses, so the total cost of distribution for x hundred sunglasses is:

TD = 30x

The total cost is the sum of the production cost and the distribution cost:

Total Cost = TC + TD = 80x + 30x = 110x

The profit is the difference between the total revenue and the total cost:

Profit = TR - Total Cost = xy - 110x

To maximize the profit, we need to find the values of x and y that maximize the profit equation. Taking the derivative of the profit equation with respect to x and y yields:

d(Profit)/dx = y - 110
d(Profit)/dy = x

Setting these derivatives equal to zero and solving for x and y gives:

y = 110
x = MR/MC = 110/80 = 1.375

Therefore, the profit-maximizing wholesale price is $110 per unit, and Slick Shades is producing 1.375 hundred sunglasses (or 137.5 sunglasses).

To find the retail price, we need to add the distributor's markup to the wholesale price. Let's assume that the distributor's markup is a percentage of the wholesale price, denoted by m. Then the retail price is given by:

Retail Price = Wholesale Price + Distributor Markup
Retail Price = 110 + m(110)

To maximize the profit, Slick Shades should choose the value of m that maximizes the profit (assuming that the distributor's markup does not affect the demand for the sunglasses). The profit equation is:

Profit = (Retail Price - 80) * 1.375 * 100

Taking the derivative of the profit equation with respect to m and setting it equal to zero gives:

d(Profit)/dm = 137.5 - 110 / (1+m)^2 = 0

Solving for m yields:

m = 0.575

Therefore, the retail price is:

Retail Price = 110 + 0.575(110) = $170

So the answer is (C).

Slick Shades Has a Constant Marginal Cost of Production Equal