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Problem 1. Monopoly (a) A monopolist faces the demand curve 𝐷(𝑝) = 100 − 2𝑝. Its cost function is 𝑐(𝑦) = 2𝑦. What is its profits? (b) Now, it faces 𝐷(𝑃) = 10𝑝−3 and 𝑐(𝑦) = 2𝑦. Find the demand price elasticity, and use that to find its optimal level of output and price. [Recall the markup definition] (c) If 𝐷(𝑝) = 100/𝑝 and 𝑐(𝑦) = 𝑦2, what is the optimal level of output? [Check this one carefully] (d) Looking again at equations (5) and (6) in the lecture, what hap- pens when the demand curve is perfectly inelastic? Does this look familiar? (e) True or false - if we impose a quantity tax on a monopolist, the market price will always increase by the amount of the tax.

          Problem 1. Monopoly
(a) A monopolist faces the demand curve 𝐷(𝑝) = 100 − 2𝑝. Its cost
function is 𝑐(𝑦) = 2𝑦. What is its profits?
(b) Now, it faces 𝐷(𝑃) = 10𝑝−3 and 𝑐(𝑦) = 2𝑦. Find the demand price elasticity, and use that to find its optimal level of output and price. [Recall the markup definition]
(c) If 𝐷(𝑝) = 100/𝑝 and 𝑐(𝑦) = 𝑦2, what is the optimal level of output? [Check this one carefully]
(d) Looking again at equations (5) and (6) in the lecture, what hap- pens when the demand curve is perfectly inelastic? Does this look familiar?
(e) True or false - if we impose a quantity tax on a monopolist, the market price will always increase by the amount of the tax.

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Added by Megan B.

Principles of Economics

Gregory Mankiw 8th Edition

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Solved by Expert Azat Nurmukhametov

09/26/2022

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Its cost function is \( c(y) = 2y \). What is its profits? ** Show more…

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Problem 1. Monopoly (a) A monopolist faces the demand curve 𝐷(𝑝) = 100 − 2𝑝. Its cost function is 𝑐(𝑦) = 2𝑦. What is its profits? (b) Now, it faces 𝐷(𝑃) = 10𝑝−3 and 𝑐(𝑦) = 2𝑦. Find the demand price elasticity, and use that to find its optimal level of output and price. [Recall the markup definition] (c) If 𝐷(𝑝) = 100/𝑝 and 𝑐(𝑦) = 𝑦2, what is the optimal level of output? [Check this one carefully] (d) Looking again at equations (5) and (6) in the lecture, what hap- pens when the demand curve is perfectly inelastic? Does this look familiar? (e) True or false - if we impose a quantity tax on a monopolist, the market price will always increase by the amount of the tax.

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Transcript

00:01 Hello, first of all, before starting doing this part of this question, let me find the total revenue.

00:12 Total revenue function, it will be very useful.

00:15 We multiply a price by quantity, and we have 300 times quantity minus 3 times quantity squared.

00:29 And total cost function is given.

00:34 So for the first question a we need to find the price quantity combination that maximize our firm's profit.

00:44 So here we use the maximization rule marginal revenue should be equal to marginal cost and we can find marginal revenue it will be the derivative of the total revenue with respect to quantity.

01:02 So marginal revenue would be 300 minus 6 quantity and marginal cost will be 4 times quantity.

01:17 And from here we can easily find it's a very simple equation.

01:22 Optimal quantity is 30.

01:26 Optimal price is 210.

01:32 This is the price quantity combination that maximizes the profit of this company.

01:43 Okay, part b.

01:45 Now let's calculate this maximized profit.

01:50 So we have both functions, total revenue minus total cost, when quantity is 30.

02:03 And now we can find our answer.

02:08 So we multiply price by quantity and subtract the total cost which is 1 ,500 minus 2 times quantity 2 times 30 squared and from here we can find that the maximum profit will be $3 ,000...

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