If D(p) = 100/p and c(y) = y2, what is the optimal level of output of
the monopolist? (Be careful.)
To determine the optimal level of output for a monopolist, we need to find the quantity that maximizes the monopolist's profit. Profit maximization occurs when marginal revenue (MR) equals marginal cost (MC). However, since we only have information about the demand function (D(p)) and the cost function (c(y)), we need to derive the marginal revenue and marginal cost functions.
Let's start by finding the marginal revenue (MR) function. The marginal revenue is the change in total revenue resulting from a one-unit increase in output. To find MR, we can differentiate the total revenue function.
The total revenue (TR) is given by the product of price (p) and quantity (q):
TR = p * q
To find the marginal revenue (MR), we differentiate TR with respect to q:
MR = d(TR)/d(q)
Since D(p) gives us the relationship between price (p) and quantity (q), we can express q in terms of p:
q = D(p)
Substituting q into the TR function:
TR = p * D(p)
Now, differentiating TR with respect to p will give us MR:
MR = d(TR)/d(p) = d(p * D(p))/d(p)
To find MR, we differentiate the expression p * D(p) with respect to p by treating D(p) as a function of p:
MR = D(p) + p * d(D(p))/d(p)
Next, let's find the marginal cost (MC) function. The cost function c(y) is given as y^2, where y represents the quantity. To find MC, we differentiate the cost function with respect to y:
MC = d(c(y))/d(y) = d(y^2)/d(y)
MC = 2y
Now that we have both the MR and MC functions, we can set them equal to each other to determine the optimal level of output.
D(p) + p * d(D(p))/d(p) = 2y
However, without a specific functional form for D(p), we cannot directly calculate the optimal output level. To find the optimal level, we need additional information about the demand function, such as a specific equation or parameters.