An monopoly exists when an industry contains only a single firm. If a firm can drive out competition because its costs of production are lower, it is a natural monopoly. Many monopolies do not fall into this category; they may be government-awarded, city-granted, or based upon actions by a vital industry (i.e. banks not financing competition). In this turn from perfect competition, the monopoly is a price-maker.  Sometimes this behavior can be engaged in by a cartel. Furthermore, offering unique products often leads to monopolistic competition. 

8.1  Monopolist's Profit-Maximizing Optimum

Price-Quantity Solution

The monopolist's profit-maximizing optimum is shown below:
Here on the left, maximum profit ( \(\Pi^*\)) occurs where the vertical difference between the total revenue curve and the total cost curve is the greatest. This is where the slopes of the lines are equal. On the right, the marginal revenue and marginal cost curves intersect out the output level. Profit is the product of the quantity and difference between average cost and price: \(Q^* \cdot (AC^* - P^*)\).
A monopolist firm can choose either product price or industry output-- not both. Once price is set, the market demand function determines the quantity that can be sold at that price, and conversely for the output. Notice that although the monopolist's optimal quantity is at the output where MC and MR intersect, the optimal price is not at that height, rather, it is higher along the industry demand curve,  \(AR \equiv D\)
Let's recall the relevant functions for each item of calculation. Marginal revenue is not the price charged for the last unit sold (marginal revenue), rather it is the profit (price - cost) from the last unit:
\(MC \equiv P + Q \frac{\Delta P}{\Delta Q}\)
As before, profit is defined as:
\(\Pi \equiv R - C \equiv PQ - C\)
The Monopolist's Maximum-Profit Condition is:
\(MC = MR \equiv P + Q \frac{\Delta P}{\Delta Q}\)
As before, there are two qualifications, 1) the MC curve must cut the MR curve from below, 2) the no-shutdown conditions must hold, where the firm can only produce if \(P > AVC\) in the short run and \(P > AC\) in the long run. 
Price elasticity demand in this scenario is given by:
\(\eta \equiv \frac{\frac{\Delta Q}{Q}}{\frac{\Delta P}{P}} \equiv \frac{\Delta Q}{\Delta P} \frac{P}{Q}\)
This leads to an expression connecting MR with the price elasticity:
\(MR \equiv P(1 + \frac{1}{\eta})\)
Since elasticity is ordinarily negative, we see again that MR is less than price. Recall that elasticity generally varies along the demand curve; in fact, it is infinite at y-intercepts and zero at x-intercepts. When demand is inelastic, MR is negative. Thus given the profit-maximizing conditions of the monopolist, he will never produce in a region of inelastic demand. 
Proposition 1: A profit-maximizing monopoly will always choose a price-quantity solution in the range of elastic demand along the market demand curve. 
Proposition 2: Given any linear demand curve \(P = A - BQ\), marginal revenue is \(MR = A - 2BQ\). The MR curve starts at the P-intercept and then falls twice as fast as the demand curve.

Monopoly vs. Competitive Solutions

The cost data of a competitive and monopolistic firm will look identical. However, instead of operating where MC = P, the monopolist will maximize profit by producing at MC = MR, which is usually a lower quantity. 
Proposition 3: The monopoly output solution occurs where marginal cost is equal to marginal revenue. Since competitive firms produce where marginal cost is equal to price, and since marginal revenue is less than price, a monopolized industry charges a higher price and produces a smaller output than a competitive industry with the same cost and demand functions. 
One last important qualification must be made. A monopolist may want to deter entry by setting a price higher than the minimum average cost of the lowest-cost potential entrant. The threat of entry may prevail over profit maximization. 

8.2 Monopoly and Economic Efficiency

The higher prices and lower output of monopolies, of course, make consumers worse off but make the producer better off. What portion is the efficiency loss? See the graph below:
If there are no productive losses or gains from organizing the industry into a single firm, the supply curve would be the same as the marginal cost curve of the monopolist. In comparison with the competitive outcome, the shaded area is the transfer from consumers to the monopolist (equal to the price difference times the quantity still produced). There is also an efficiency loss from producer and consumer surplus because of reduced trade. 

8.3 Regulation of Monopoly

In the US, privately owned public utilities such as electricity and water (thought to be "natural monopolies") are commonly regulated. This regulation aims to limit the monopolist to a normal accounting profit, just adequate enough to attract the needed capital and other resources into the business. This corresponds to zero economic profit,  thus aiming to achieve the same result had competition been possible. Such a regulation (increasing costs) can be drawn like this:
Output can be excessive because resources have alternate uses; thus regulation aims at preserving efficiency. However, while the unregulated monopoly produces too little, the regulated monopoly produces too much and so it is hard to say which is more efficient. 
In conclusion, if the average cost curve is rising in the relevant range, the regulatory zero-profit solution increases output beyond the monopolist's profit-maximizing solution. Such regulation is inefficient. With a falling average cost curve, the regulatory zero-profit solution increases output insufficiently. 

8.4  Monopolistic Price Discrimination

Sometimes a firm may use discriminatory pricing schemes to divide the market and offer different prices so as to further exploit customers. 

Market Segmentation

Dividing up the market can prove lucrative. For example, Japan auto manufacturers have been accused of "dumping," or charging lower prices abroad than in their country since demand is arguably more elastic on the huge world market. The figure below pictures a manufacturer who can charge two different prices in two different markets (with two separate marginal revenue curves):
The firm will want to set the marginal cost equal to each of the marginal revenue curves: 
\(MC = mr_1 = mr_2\)
Knowing that the two marginal revenues are equal, it follows that:
\(P_1(1 + \frac{1}{\eta _1}) = P_2(1 + \frac{1}{\eta _2})\)
If demand is more elastic in one market, the profit-maximizing monopolist will set the lower price there. Examples of this in practice are such things like: discounts for elderly and children, off-season pricing, discount coupons and lunch specials.  

Block Pricing & Perfect Discrimination

In a block pricing scheme, the seller charges a different price to a single consumer, for example, a discount when purchasing a greater quantity. This two-part pricing scheme allows the monopolist to capture the portion of consumer surplus below the price he charged for one unit. An illustration of this:
Although this scheme might seem very common, there are also good reasons for such pricing, as the lump-sum fixed cost is often high for the first set of units and falls thereafter. 
Perfect price discrimination is efficient, since the monopolist is charging a price equal to the marginal cost at the lowest price offered. This means that each buyer's marginal value (demand price) is equal to the seller's cost. 

8.5 Cartels

A cartel is a group of firms behaving as a collective monopoly. Cartels have an Achilles heel, however, since it  always pays for a single firm to cheat. A cartel can raise price over the competitive level only by reducing aggregate industry output. But at higher prices, a member firm can profit by covertly producing even more than at the competitive equilibrium, and so can nonmembers. The added production can usually subvert the cartel. 
Cartels are illegal, thus they must be effective and secret (unlikely combination) to be viable. It is interesting to note that the Webb-Pomerene Act allows American exporters to form a cartel for dealings in foreign markets. One of the most famous cartels is the Organization of Petroleum Exporting Countries (OPEC). The members of the cartel were sovereign governments rather than private firms, though its power has declined due to chiseling and increased production by nonmembers. 

8.6 Network Externalities

A person's demand for a good is usually independent of how much others buy. However, sometimes a good is more valued when more people by it, thus exhibiting positive network externalities. This happens often with technology. 

Demand for a Network Good

We cannot simply find the aggregate demand curve by horizontally summing the the individuals' demand curves as before. Below, we can see what the demand curve for the Internet might look like:
Customers expect the value of the good to rise with each additional user. To find market demand, we assume these expectations are correct. This creates a flatter demand curve, meaning a positive network effect increases the responsiveness of demand to price changes. Furthermore, there is also a "chicken and egg" startup problem since the choke prices (highest demand) might be lower than the costs of production.
Network effects may also cause natural monopolies, but for reasons on the demand side. An important source of network externalities is the convenience of a common standard or format. 

The Lock-in Issue

Network effects might possibly lock an industry into an inferior technology. The cost of switching just might be too high at the outset. Market forces at work, however, tend to overcome lock-ins since the demand will eventually swell to a sufficient size. 
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