A monopolist operates with the following data on cost and...

A monopolist operates with the following data on cost and demand. It has a total fixed cost of $1000 and a total variable cost of Q^2, where Q is the number of units of output it produces. The firm's demand function is Qd = 60-0.5p. The firm expects the conditions of demand and cost to continue in the foreseeable future. a) What is the inverse market demand curve? b) What is the marginal revenue function that corresponds to the market demand curve? c) What is the monopolist's marginal cost function? d) What is the firm's revenue if it operates and maximizes profit? e) What is the deadweight loss from the monopoly pricing? f) Should the firm continue to operate in the short run, or should it shut down?

Transcript

00:01 So here we have a monopoly.

00:02 We have a fixed cost of a thousand.

00:05 We have a variable cost, total variable cost of q squared.

00:10 I believe that's q squared.

00:12 And we have a demand curve.

00:13 Quantity demand that is 60 minus 0 .5p.

00:17 The firm expects the contingency.

00:20 So first of all, the inverse is to rewrite in terms of p, right? rewrite for p.

00:26 That's what we mean by inverse.

00:28 So i could rearrange this, is 0 .5p is 60 minus q or p is equal to 120 minus 2 q.

00:37 That would be my inverse demand for a curve.

00:40 B, i want to think first about revenue.

00:43 Revenue is price times quantity.

00:45 I can now sub in that inverse demand curve for p.

00:50 And now i can get marginal revenue, which is the derivative of revenue with respect to quantity, which is 120 minus 4q.

00:57 That would be the marginal demand curve.

01:02 C, marginal cost is the derivative of total cost with respect to quantity, and that's just equal to 2q, right? total cost is now q squared plus 1 ,000, right? it's the two costs added together, but when you differentiate 1 ,000, it just goes.

01:20 So that would be the marginal cost.

01:22 So for d, we want to think about putting marginal revenue equal.

01:27 To marginal cost.

01:29 That defines the firm's maximum profit, right? so if we do that, i get 120 minus 4q is equal to 2q, 120 is equal to 6q, q is equal to 20, and q is equal to 20 will result in a price of 120 minus 2 times 20, which is equal to a price of 80, right? that would be the firm's maximum the profit maximizing choice.

02:01 The revenue is equal to price times quantity, which is going to be 20 times 80, which is equal to 1 ,600.

02:09 And that would be the revenue at the optimal outcome.

02:13 Deadweight loss, i want to visualize.

02:15 So if we think of dead weight loss, right, for e, dead weight loss is demand.

02:22 We have marginal revenue.

02:24 We have marginal cost...