# How to Solve Monopoly Markets (linear Equations)

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Say you're given a monopoly market to solve:

A monopolist has a demand curve given by D: P = 100 - Q and a marginal cost curve given by S: P = 2Q.

How would you solve this?

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## Step 1: Graph the Market

Plot supply and demand with P on the vertical axis and Q on the horizontal axis. Notice that in the monopoly case, supply is marginal cost.

## Step 2: Derive Marginal Revenue

Without competition in the market, a monopolist doesn't produce where S=D. Instead, he wants to maximize his marginal revenue. With linear demand, marginal revenue has the same intercept as demand, but twice the slope. (For those with a calculus background, this is because total revenue is demand (equal to P) times Q, and then take the derivative with respect to Q). This gives us MR=100-2Q.

## Step 3: Finding P and Q

So where will the monopolist produce? Where MR=MC, our golden rule for maximizing profit. However, this only determines Q. To find P, we substitute that Q back into demand to find P.

In other words, the monopolist chooses Q to maximize TR, and charges "as much as he can get away with"--the highest price consumers will pay for that profit-maximizing Q.

## Step 4: Comparing Efficiency

The deadweight loss from this market being controlled by a monopolist is the difference in total surplus between the monopoly situation and the point of social efficiency (where supply--MC--equals demand).

The orange area represents consumer surplus under monopoly, the purple area represents producer surplus under monopoly, and the light green area represents deadweight loss. You can easily see that at the socially efficient point, some of producer surplus and DWL would be allocated to consumers, and the rest of DWL would be allocated to producers.

## Step 5: Calculating DWL Precisely

As deadweight loss is a triangle, we calculate it as 1/2*b*h.

DWL=.5*(33.3-25)*25=104.16

You could also calculate this as the change in total surplus, calculating the sum of producer and consumer surplus under monopoly and competition.

**Note that the 104.16 is calculated using 33.33333 (repeating) rather than 33.3. If you use 33.3, you will get 103.75, which is also acceptable.

## Discussions

Question 3 months ago on Introduction

How to find quantity