If p = 20 - y and TC(y)= 8y, then a profit- maximizing monopolist will set a price equal to:
A)8.
B)7.
C)12.
D)14.

Question

Quizplus · Accepted Answer

To find the profit-maximizing price, we need to first find the monopolist's profit function, which is given by:

π(y) = p(y) * q(y) - TC(y)

where q(y) is the monopolist's demand function, which we can find by solving for y in the inverse demand function:

p(y) = 20 - y

rearranging this, we get:

y = 20 - p(y)

Substituting this into the TC function, we get:

TC(y) = 8(20 - p(y))

which we can simplify to:

TC(y) = 160 - 8p(y)

Substituting these expressions for p(y) and TC(y) into the profit function, we get:

π(y) = (20 - y) * y - (160 - 8p(y))

Expanding this, we get:

π(y) = 20y - y^2 - 160 + 8p(y)

Taking the derivative of π(y) with respect to y and setting it equal to zero, we get:

dπ(y) / dy = 20 - 2y + 8dp(y) / dy = 0

Solving for p(y), we get:

p(y) = (20 - y) / 2

Substituting y = 10 into this expression, we get:

p(y) = (20 - 10) / 2 = 5

However, this is the marginal cost, not the price that maximizes profits. To find the profit-maximizing price, we need to substitute this value of p(y) into the demand function to get:

q(y) = 20 - 5 = 15

Therefore, the profit-maximizing price is:

p(y) = 20 - q(y) = 20 - 15 = 5

So, the correct answer is not one of the choices given. However, if we consider the closest option to this answer, it would be D) 14, which is 9 units above the profit-maximizing price of 5. Therefore, D is the closest option, but not the correct one.

If P = 20 - Y and TC(y)= 8y, Then