# Monopoly Price and Output

###### Formerly at hspm.sph.sc.edu/ECON/Monopoly/Mon.html

This tutorial shows how, in theory, a business firm that monopolizes its industry finds the price and output rate that maximize profit.

A "monopoly," in economics jargon, is the sole seller of its product.  The way this is represented mathematically is to give the firm a downward sloping demand curve.  This means that a monopoly is a firm whose demand is not perfectly elastic.  The monopolist can change the price, and the result is that the amount sold changes.  This is contrast to a competitive firm, which must sell at the going price, take it or leave it, but can sell an unlimited amount of product at that price.

The monopolist is a "price maker," not a "price taker."

This may seem like a too broad a definition of monopoly.  Is Wendy's, the fast food chain, a monopoly?  Their demand is not perfectly elastic -- they can raise their prices without having the quantity demanded drop to zero. That makes them a monopoly, yes?

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```

Both answers are correct.  Wendy's is not a monopolist, in any sense that would get it in trouble with the law, but it is the only seller of its particular brands and formulations.  Economists call this kind of market "monopolistic competition," a term coined by economist Joan Robinson.  Each firm that sells fast food has a demand curve with some slope; they can sell more by lowering their prices, or sell less by raising their prices.  This means that the monopoly analysis can be applied.  But the demand curve is not very inelastic, so no great excess profits can be made, because other firms' fast food can nearly substitute for Wendy's.

 Market Type What the Firm's Demand Is Like Pure competition Elastic demand.  The firm can't raise its price above the market price and expect to sell any.  The firm doesn't lower its price below the market price, because it can sell all it wants at the market price, so why ask for less? Monopolistic competition Demand is not perfectly elastic.  If the firm raises its price, quantity demanded does not go down to 0.  If the firm lowers its price, quantity demanded increases, so it is possible that a price cut might increase revenue. Monopoly Demand is not perfectly elastic.  If the firm raises its price, quantity demanded does not go down to 0.  If the firm lowers its price, quantity demanded increases, so it is possible that a price cut might increase revenue.

Is the distinction between monopolistic competition and monopoly clear to you?

Most businesses we encounter in the course of a day are considered monopolistic competitors.  They are in markets that have product differentiation, meaning that buyers can tell the difference between one seller's version of the product and another seller's version. The seller's market power is limited, however.  In monopolistic competition, no great public harm results if one seller raises price above the competitive level.

Medical practices, for example, have product differentiation.  People recognize differences among doctors, even though the differences are often a matter of style, and not of measurable quality.  A physician's demand is usually not completely elastic.

The line between monopoly and monopolistic competition is a matter of debate.  Antitrust cases can turn on where the judge puts the distinction.  The issue is typically phrased in terms of what market is. For example, in the aluminum antitrust case of 1946, the Aluminum Company of America (ALCOA) argued that its market was metals, and that its aluminum competed with steel in the monopolistic competition sense.  The judge decided that aluminum was different enough from other metals to be a market of its own, and that ALCOA had a near-monopoly of that market, which it did in those days, selling 90% of new aluminum in North America. (ALCOA also argued that the market should include scrap aluminum, as well as new aluminum. The judge rejected that, too, on the grounds that all the scrap aluminum used to be new.)

One possible distinction would be by the elasticity of demand at a competitive price level.  I might say that monopolistic competition is characterized by elastic demand at competitive price levels, while monopolists have inelastic demand at competitive price levels.  By "competitive price level," I mean what the price would be if the market were competitive.  This qualifying phrase has to be added because an unrestrained profit-maximizing monopoly will raise its price to where its demand is elastic, as the following analysis shows.  That is why, to use elasticity of demand as a criterion, you have to ascertain what the price would be if the market were competitive.

This next section of the tutorial shows how monopoly (or monopolistic competition) pricing decisions can be analyzed.  The analysis is based on the assumption that the firm seeks to maximize its profit.  Real-life firms can have other goals.  This analysis shows what is theoretically possible.  It leads to the diagram, that you can find in almost any introductory economics textbook, that shows a demand curve, a marginal revenue curve, and a marginal cost curve.  The profit-maximizing output rate is where the marginal revenue curve and the marginal cost curve intersect.

## Marginal Revenue

The concept we'll need to analyze monopoly pricing and output decisions is "marginal revenue."  We'll develop the marginal revenue concept with some numerical examples.

### Marginal Revenue Defined

Marginal revenue is defined as the addition to total revenue that comes increasing by one unit the rate at which you sell your product or service.

You might think that the additional revenue you'd get from selling one more unit per day (or whatever time period you want) would just be the price at which you sell the extra unit.  That's true if you are a price-taker, but not if you are a price-maker.

If you are a price-taker, and you want to sell one more unit per day, you just sell it.  Your demand is elastic and the market will take all you want at the going price.

If you are a price-maker, and you want to sell one more unit per day, you must lower your price.  This makes the calculation of how much revenue you gain more complex.  You get some additional revenue by increasing the rate of your sales, but you also lose some revenue because you are getting less money for each item sold.  For this reason, your marginal revenue less than your price.  It is even possible for the marginal revenue to be negative.  This happens if what you lose from lowering the per-item price is more than what you gain from selling additional items.

Not yet clear?  Don't worry.  Let's go through an illustration.  We start with the simpler case, the case of the price-taking firm.

### Marginal Revenue in the Price Taker (Horizontal Demand Curve) Case

Imagine that you run a primary care clinic.  Your "product" is visits.  If there is stiff price competition among clinics in your area, your demand may look something like this:
```Price per visit:
\$   20    20    20    20    20    20    20    20    20    20

Quantity of visits demanded per hour:
0     1     2     3     4     5     6     7     8     9
```
This is a peculiar demand table.  All the numbers in the price row are \$20!  What about higher and lower prices?  At prices above \$20, you sell nothing.  At prices below \$20, you can sell as much as you want, but you'd just be throwing
money away, because you can sell as much as you want at \$20.  Such a demand table is characteristic of "competitive" markets in which no one firm can influence the price.

Here's that demand table translated into a graph.  Every price-quantity pair in the table corresponds to a point "*" on the graph.

#### Horizontal Demand Curve -- Firm Is Price Taker

```PRICE

100
90
80
70
60
50
40
30
20  *     *     *     *     *     *     *     *     *     *
10
0
0     1     2     3     4     5     6     7     8     9  QUANTITY
DEMANDED
```
The points "*" form a horizontal line, which is why this is called a horizontal demand curve.

If our quantities were in hundreds or thousands, then instead of ten discreet points, we'd have a horizontal line.

#### Horizontal Demand Curve -- Firm Is Price Taker -- Continuously Variable Output Rate

```PRICE
100
90
80
70
60
50
40
30
20  -------------------------------------------------------
10
0
0     1     2     3     4     5     6     7     8     9    QUANTITY
DEMANDED
in millions
```

### Total Revenue

As a first step towards calculating the marginal revenue, let's calculate the total revenue at each output rate.

Definition:  Total Revenue = Price times Quantity.

More precisely, Total Revenue, in dollars per hour, equals the Price of each unit multiplied by the number of units sold per hour. (The time period doesn't have to be hours; it can be days or years or any unit of time.  The currency unit doesn't have to be dollars; it can be any form of money.)

Let's calculate some Total Revenue numbers for this price-taking firm.  Click on a text field below (the little boxes are called "text fields") and type in the Total Revenue for that output rate. Type a plain number with no \$ sign. Press Enter and I'll tell you if you were right.

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```

Total revenue is proportional to how much you sell, if the firm is a price-taker.

Now let's calculate the ...

### Marginal Revenue for Price-Taking Firm

The marginal revenue is the change in total revenue from going from one output rate to the next.  In the applet below, click on a text field and type in the appropriate marginal revenue. (Type a plain number with no \$ sign.)
Press Enter and I'll tell you if the number you typed was right.

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```

Pretty boring, huh?  All the marginal revenues are the same.  The confirms that, for the price taking firm, marginal revenue equals the going price, which is \$20 per visit in this example.

Let's see this on a graph:

#### Demand and Marginal Revenue -- Price Taker Firm

The  m's  on the graph show marginal revenue.  The first  m  on the left,
for example, means that the marginal revenue from increasing sales from
0 visits per hour to 1 is \$20.

```        * = Demand Curve Points
m = Marginal Revenue Curve Points
PRICE
100
90
80
70
60
50
40
30
20    * m * m * m * m * m * m * m * m * m *
10
0
0   1   2   3   4   5   6   7   8   9    QUANTITY DEMANDED
```
If there were a very large number of points, instead of just the 10 discrete output levels we have here, the points would blend together into smooth lines.  The demand curve and the marginal revenue curve would coincide.

To sum up: If the price is fixed by the
market or by government, and demand is elastic (you can sell all you want at the going price) then
marginal revenue = the price.

Let's look at the Total and Marginal Revenues again.

```Price per visit:
\$    20     20     20     20     20     20     20     20     20     20
Quantity of visits demanded per hour:
0      1      2      3      4      5      6      7      8      9
Total Revenue per hour (=Price times Quantity):
\$     0     20     40     60     80    100    120    140    160    180
Marginal Revenue (= difference in Total Revenue)
\$        20     20     20     20     20     20     20     20     20
```
The preceding interactive tutorial said that the marginal decision rule for setting output rate for the competitive firm was: Increase the output rate so long the price received is greater than the marginal cost.

We can now restate the marginal decision rule more generally: Expand production so long as marginal revenue exceeds marginal cost.

To reiterate the explanation: The marginal revenue is what you gain if you sell one more visit. The marginal cost is what it costs if you sell one more visit. If selling one more visit per hour gains you more than it costs, a "rational" decision maker sells one more visit.

```Price per visit:
\$    20     20     20     20     20     20     20     20     20     20
Quantity of visits demanded per hour:
0      1      2      3      4      5      6      7      8      9
Total Revenue per hour (=Price times Quantity):
\$     0     20     40     60     80    100    120    140    160    180
Marginal Revenue (= difference in Total Revenue)
\$        20     20     20     20     20     20     20     20     20
Marginal Cost
\$        15     15     15     15     15     15     15     25     30
```
See if you can apply the marginal decision rule to answer this:
Based on these marginal cost and marginal revenue figures, what is the profit-maximizing quantity?

(Click here to scroll up to see the whole table that you need to answer this question.)

### A Price Maker's Demand Curve

Now we're ready to compare the price maker situation. To do this, we just give the firm's demand curve a downward slope. Now our firm is no longer a passive price taker. It can choose what price to charge. Sales go down if the price goes up, and vice-versa.
```                    DEMAND TABLE -- Price Maker (Monopoly)

Price per visit:
\$   100     90     80     70     60     50     40     30     20     10

Quantity of visits demanded per hour:
0      1      2      3      4      5      6      7      8      9
```
In economic theory, any price maker is, is effect, a monopolist. Suppose yours is the only clinic in your area. You don't have to charge the going price because there is no going price. You can charge what you want, but what you charge determines how many visits you get. To get more patients you must charge less. Raise your price, and you get fewer patients. Your happy problem is to jack up the price to the optimal level. You can find the optimal level by using the marginal decision rule. To do this you must calculate the marginal revenue. But first, let's see a graph of this demand table.

Here's the demand table translated into a graph. Each price-quantity pair in the table corresponds to a point "*" on the graph.

```            Demand Curve -- Price Maker

PRICE

100    *
90        *
80            *
70                *
60                    *
50                        *
40                            *
30                                *
20                                    *
10                                        *
0
0   1   2   3   4   5   6   7   8   9    QUANTITY DEMANDED
```
The points "*" form a descending line in this price maker case.

Before we go further I need to point out a crucial hidden assumption, namely that at any given price, you sell as much as people will buy at that price.

In reality, most businesses decide both the prices they want to charge and the quantities they'll produce for sale. For example, here you could decide to charge \$40 per visit, but only see 3 patients per hour. There would be excess demand, because at that price people would want 6 visits, but you could turn the extras away or make them wait.

The crucial assumption is that you do not turn sales away. Instead you always charge what the market will bear. If you want 3 visits, you charge no less than \$70 each. If you want to charge \$40 you expand your capacity to accommodate the demanded quantity, which is 6.

This implies, for example, that if you are providing 4 visits per hour, you must be charging what price? (Here's the demand table again, for reference.)

```Price per visit:
\$   100     90     80     70     60     50     40     30     20     10
Quantity of visits demanded per hour:
0      1      2      3      4      5      6      7      8      9
```

And the only way to expand service to 5 visits per hour would be to drop the price to what?

#### Price-Maker's (Monopolist's) Total Revenue

The first step to getting marginal revenue is getting total revenue, as before.  The total revenues form a pattern I'd like you to notice, so please fill in the blanks in this table:

```

```

In the monopoly case, total revenue goes up and then goes down, as the output rate rises. This is unlike the competitive firm, for which the total revenue goes up linearly as the output rate rises.  That's because you have to charge lower and lower prices to sell more and more.

Let's bring in the elasticity concept. Elasticity is discussed in an another interactive tutorial. We'll use "elastic" here to mean that the elasticity of demand has an absolute value greater than 1. We'll use "inelastic" here to mean that the elasticity of demand has an absolute value between 0 and 1. ("Absolute value" means changing negative numbers to positive.)

Here again is a table showing prices, quantities, and total revenue.

```Price per visit:
\$   100     90     80     70     60     50     40     30     20     10
Quantity of visits demanded per hour:
0      1      2      3      4      5      6      7      8      9
Total Revenue:
\$     0     90    160    210    240    250    240    210    160     90
```
Over which range of output rates is the demand elastic?

```

```

```Price per visit:
\$   100     90     80     70     60     50     40     30     20     10
Quantity of visits demanded per hour:
0      1      2      3      4      5      6      7      8      9
Total Revenue:
\$     0     90    160    210    240    250    240    210    160     90
<<<<<<<<<<<<<<<<<<Rising revenue<<<<<<<<<>>>>>>>>>Falling revenue>>>>>
<<<<<<<<<<<<<<<<<<Elastic demand<<<<<<<<<>>>>>>>>>Inelastic demand>>>>
```
At high prices, a monopolist's demand is elastic, responsive to price changes.  At low prices, a monopolist's demand is inelastic, not very responsive to price changes.  In practice, what is a "low price" and what is a "high price" depend on consumers' budgets and on the prices of other things consumers can buy to help meet their need for this firm's product.

Here is an animated display of the relationship between price and total revenue.
This diagram has a continuous-looking demand curve, unlike the discreet demand relation shown in the table above. Imagine for now that we are a large organization, so that each unit of quantity in the table above actually represents a large number of visits, so that fractional numbers of visits make sense.

If we had no costs, and all we cared about was maximizing revenue, then we would set the price at \$50. We would see 5 patients per hour, and make \$250 per hour.

### A Monopolist's Marginal Revenue

The marginal revenue is the change in total revenue resulting from a change of one unit in the output rate.  In the bottom row of the applet below, click on a text field and type in the appropriate marginal revenue. (As always, type a plain number with no \$ sign.) Press Enter and I'll tell you if that marginal revenue is right. To be sure you see the pattern, I'll be asking you do fill in most of the text fields on the bottom row.

```

```

Marginal revenue goes down as the quantity you try to sell increases. If, as here, the demand curve is inelastic in some price range, the marginal revenue is negative in that price range.  Negative marginal revenue means that you lose revenue if you lower price and expand sales.  A negative marginal revenue implies that if you reverse you price cuts and raise prices, your sales shrink but your total revenue goes up.

In view of that, try this True-False question:

Uwe Reinhardt, in "Perspective: Our Obsessive Quest to Gut the Hospital," Health Affairs, Summer 1996, 15(2), pp. 145-154, argues that hospitals should allocate more of their overhead costs to services whose demand is inelastic, and allocate less overhead costs to services whose demand is elastic. In effect, he is applying this principle, that the monopolist gains revenue by raising price where demand is inelastic.

Here's a graph showing our demand and marginal revenue numbers.  This graph shows the first and fourth lines of the table that you filled in above.  (Scroll up about one full screen to see it.)
In this graph, the points on the demand line are *'s; the points on the marginal revenue line are m's.

```            Demand and Marginal Revenue -- Price Maker

PRICE

100    *
90      m *
80            *
70          m     *
60                    *
50              m         *
40                            *
30                  m             *
20                                    *
10                      m                 *
0    0   1   2   3   4   5   6   7   8   9         QUANTITY DEMANDED
-10                          m
-20
-30                              m
-40
-50                                  m
-60
-70                                      m
```
The marginal revenues lie below the demand points, and fall twice as fast.  The m's go negative when the quantity demanded is above 5, which happens when the price drops below \$50.

I can show that also with an animated graph.
An apology to folks with 640x480 displays:  Unfortunately, I have to make the graph too large for your screen. Most of the action is in the top half of the graph.

Next is an illustration for a larger firm, with nearly continuously variable output. A we move along its demand line, raising and lowering price, marginal revenue traces out a line with twice the slope of the demand line.

### Marginal Revenue, Marginal Cost, and the Profit-Maximizing Output Rate

Now let's go back to our table and add marginal costs to it: (The table is getting big!)
```Price per visit:
\$   100     90     80     70     60     50     40     30     20     10
Quantity of visits demanded per hour:
0      1      2      3      4      5      6      7      8      9
Total Revenue:
\$     0     90    160    210    240    250    240    210    160     90
Marginal Revenue
\$        90     70     50     30     10    -10    -30    -50    -70
Marginal Cost
\$        15     15     15     15     15     15     15     25     30
```
To save space, I didn't include a row for Total Cost.  We don't need that to calculate the profit-maximizing output rate.  All we need is the marginal revenue, marginal cost, and the restated marginal decision rule:  Expand output so long as the marginal revenue is greater than the marginal cost.
What is our profit-maximizing output rate?

```

```

Notice that, at 4 visits, we are in the elastic portion of the demand curve.  We don't just raise our price so long as the demand is inelastic.  We cut back even more than that, so as to give us the biggest profit above our costs.

The monopolist allows 4 visits and charges \$60 for each. A competitive industry with the same costs and demand would, as we saw earlier, have 7 visits and charge \$30 each.

With monopoly, there's an allocation and a distribution issue.  The distribution issue is easier to see:  The monopolist raises the price.  This redistributes income from the costomers to the monopolist. That would be a sufficient complaint for most of us, but not for orthodox economists, who insist on not making value judgements about which people are more deserving of income.

Economists tend to emphasize the allocation issue, which we can illustrate with our example.

```Price per visit:
\$   100     90     80     70     60     50     40     30     20     10
Quantity of visits demanded per hour:
0      1      2      3      4      5      6      7      8      9
Marginal Revenue
\$        90     70     50     30     10    -10    -30    -50    -70
Marginal Cost
\$        15     15     15     15     15     15     15     25     30
```
The profit-maximizing monopolist sets the price at \$60 and has four visits per hour.  Somewhere out there, though, is a 5th person who would come in if the price were \$50.  (We know this because the first two rows tell us that if we charge \$50 we can sell 5 visits per hour.)  That visit must therefore be worth at least \$50 to that person.  The marginal cost of providing that visit is \$15.  (The fourth row of numbers tells us that.)  By not providing 5 visits, we are giving up an opportunity to turn \$15 worth of resources into \$50 worth of service.  That's a net loss to society of \$35.  We are not using our resources in the best possible way.

This is called a "consumer surplus" argument, and we can make it again with the 6th and 7th visits.  By not providing that 6th visit, we are failing to turn \$15 worth of resources into \$40 worth of service.  By not providing the 7th visit, we are failing to turn \$15 worth of resources (the marginal cost of the 7th visit) into \$30 worth of service.

A competitive market would provide the 5th, 6th, and 7th visits.  If somebody could enter our market, they could make money providing those visits, because the prices they could get would exceed the marginal costs.  A competitive market would stop at 7, because the 8th visit has a marginal cost higher than the price you get get for it.  Still, that's a lot more than the 4 visits that the monopoly finds profit-maximizing.

### Summary

Marginal revenue is the change in revenue that results from a unit change in the amount sold.  The marginal revenue curve can be mathematically derived from the demand curve.

Monopolies (or any firms with downward-sloping, non-perfectly-elastic, demand curves) maximizing their profit by raising price and reducing output so long as the demand is inelastic.  Then they raise the price even more, until the marginal revenue lost from the reduction in sales is greater than the marginal cost saved thanks to a lower rate of operation.

The monopoly's high price transfers income from customers to the monopolist.  The monopoly's low output rate wastes social resources, by not turning some resources into products that would be valued more highly than the resources are.

### Comment

Monopolies cause other problems, too.  They can have higher costs than necessary, taking their monopoly profits in the form of extra costs.  Monopolies can ignore opportunities to innovate in their own industries.  They can aggressively stifle innovation in related industries if they see a threat to their existing monopoly. All these issues go beyond the price theory that this tutorial is about, and may be more important.

That's all for now. Thanks for participating!