1. You are the manager of a monopoly, and your analysts have...

1. You are the manager of a monopoly, and your analysts have estimated your demand and cost functions as P = 600 - 3Q and C(Q) = 2,000 + 2Q^2, respectively. a. What price-quantity combination maximizes your firm's profits? b. Calculate the maximum profits. c. Calculate the maximum revenues. 2. A firm sells its product in a perfectly competitive market where other firms charge a price of $110 per unit. The firm estimates its total costs as C(Q) = 70 + 14Q + 2Q^2. a. How much output should the firm produce in the short run? b. What price should the firm charge in the short run? c. What are the firm's short-run profits?

Transcript

00:01 Okay, let me try to walk you through this problem.

00:03 Just a quick caveat, you've written the cost function as 2000 plus 2q2.

00:08 I'm pretty sure that's 2 ,000 plus 2q squared, right? if it's not 2q squared square, this question doesn't make any sense.

00:15 So the first thing we have to think about is that a monopolist controls the marketplace and therefore has pricing power, right? the monopolist wants to restrict quantity to make profit, right? drive up the price by restricting quantity.

00:34 The monopolist rule is to set marginal revenue equal to marginal cost.

00:38 That is, the monopolist should keep producing until the total revenue gained by one more unit is equal to the total cost of producing that unit, right? because that would, if marginal revenue is greater than marginal cost, it says that you should keep producing to increase.

00:54 Your profits.

00:56 So let's calculate revenue.

01:00 Revenue is equal to price times quantity, which is equal to 600 q minus 3q squared.

01:07 That means that marginal revenue or d revenue with respect to q is equal to 600 minus 6q.

01:17 Now, i don't know if you use calculus, but sometimes people say that for linear demand curves, marginal revenue is the same thing, except i've doubled the selection.

01:25 If you don't like calculus, that's all i'm doing.

01:28 I'm doubling the slope.

01:29 I'm going from 600 minus 3q to 600 minus 6q, but calculus is the reason why that happens.

01:35 And i need to get marginal cost as well.

01:37 Marginal cost is the derivative of cost with respect to q, which is equal to 4q, right? the 2000 doesn't have any q in it, so it disappears.

01:46 So i want to set these things equal to each other.

01:49 I've got 600 minus 6q equals to 4q, 600 equals to 3q, 600 equals to 10 q, which means that q is equal to 60.

01:59 How do i get the price? i plug this back into the demand curve, right? plug in.

02:07 So i get 600 minus 3 times 60, which is a price of 420.

02:14 Nice.

02:16 By plugging this back into the demand.

02:19 So there's my optimal price quantity relationship.

02:24 Now that i know the quantity and the price, i can calculate the revenue.

02:28 And the profit.

02:29 I'm going to do the revenue first because it's easier.

02:31 I have to do revenue to get the profit.

02:34 Revenue we said is equal to 600 q minus 3q squared.

02:40 I need to plug in 60, so that's 600 times 60 minus three times 60 squared.

02:46 Now it becomes a calculator question.

02:49 That gives me profits in my calculator of 20 ,000, revenues of 25 ,200.

02:55 Right? we can now think of, about profits.

03:00 Profits are revenues minus costs, right? i want to try to keep the cost function on screen here.

03:08 So profits is equal to revenue minus 2 ,000 minus 2 times 60 squared, right, which is the cost function.

03:21 So that's 25 ,200 minus 2 ,000.

03:27 Or actually, i'm just going to do this on one step...

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