If D(p) = 100/p and c(y) = y2, what is the optimal level of output of
the monopolist? (Be careful.)
To find the optimal level of output for a monopolist, we need to maximize profit. Profit is maximized when marginal revenue (MR) equals marginal cost (MC).
Given that the demand function is D(p) = 100/p and the cost function is c(y) = y2, we need to find the revenue function and the cost function first.
Revenue function (R):
Since the monopolist is selling the output level y at a price p, the revenue can be calculated by multiplying the price by the quantity sold. In this case, it is R = p * y.
However, we need to express the revenue function in terms of y, not p. To do this, we substitute p with the inverse demand function in terms of y.
To find the inverse demand function, we set the demand equation equal to p and solve for p:
p = 100/y
Now, we substitute this expression for p in the revenue equation:
R(y) = (100/y) * y
R(y) = 100
So, the revenue function is R(y) = 100.
Cost function (C):
The cost function is given as c(y) = y^2.
Now, we can find the marginal revenue (MR) and marginal cost (MC).
Marginal Revenue (MR):
Marginal revenue is the change in total revenue resulting from selling one more unit. Since the revenue function is constant, the marginal revenue is equal to the constant revenue, which is 100 in this case.
MR = 100
Marginal Cost (MC):
To find the marginal cost, we differentiate the cost function with respect to y:
MC = 2y
Now that we have both the marginal revenue and marginal cost, we need to equate them to find the optimal level of output:
MR = MC
100 = 2y
Solving for y, we get:
y = 50
Therefore, the optimal level of output for the monopolist is y = 50 units.