Strategy for Information Markets/Background/Monopoly pricing

When a firm makes decisions about output and prices, it is almost always the case that profit is maximized where marginal revenue equals marginal cost. The exact implications of this can vary depending on whether the firm is in a competitive market, and whether the firm is setting a single market price or engaging in more complicated pricing schemes.

Here we'll take a look at monopoly pricing, meaning:
 * 1) The firm has some amount of market power, and so faces a downward-sloping demand curve.
 * 2) The firm is setting a single market price--any units they sell are sold at the same price.

Market power
We'll talk about this kind of pricing as "monopoly pricing" because it's clearest in the case of a monopoly. A monopoly means the entire market is served by a single firm, and so the market demand curve is the same as that firm's demand curve. However, all that is required for this kind of analysis is that the firm's demand curve slopes downward, even if only slightly.

A downward-sloping demand curve means that the firm is not a price-taker. So, when the firm raises its price, it will sell some amount less, and when it lowers its price it will sell some amount more. That's in contrast to a competitive market where a firm which raised its price above the market equilibrium would sell nothing at all, and if it lowers its price below market equilibrium, it will be losing money.

Marginal revenue / marginal cost condition
In almost all cases, whether an industry is competitive or monopolistic, a firm can maximize its profit by selling the quantity for which marginal revenue (MR) equals marginal cost (MC). The intuition for that result is as follows:

Suppose that MR was greater than MC. That would mean the revenue from selling an additional unit is greater than the cost of making that additional unit. As long as this is the case, the firm will continue to produce more units, since each unit is profitable. Suppose instead that MR was less than MC. That would mean the cost of making an additional unit is greater than the revenue from selling it. If this is the case, the firm will cut back on its production, since it is taking a loss on those units.
 * when MR < MC, the firm is losing money, so it will produce less.
 * When MR > MC, the firm is making money, so they will want to produce more.
 * when MR = MC, this is an optimum level where the firm is making the most profit.

The MR=MC profit-maximizing condition applies to a firm in a competitive market, but in a trivial way. A Since a firm in a competitive market can't influence the price (p), MR = p regardless of how much output the firm produces. Therefore, a firm in a competitive market will maximize its profit by choosing a level of output where MC = p.

Since a monopoly has a downward sloping demand curve, they can influence the price. If they want to sell more, they have to lower the price they charge. That affects their revenue in two ways.

Suppose that the monopolist first is selling $$Q_1$$ units at a price of $$p_1$$. Then they lower their price from $$p_1$$ to $$p_2$$, and sell one more unit because of the lower price. The firm's revenue has increased by $$p_2$$ because they sold the extra unit at that price. However, revenue has also decreased by $$(p_1 - p_2)Q_1$$ because they are selling all the other units at the lower price as well. The second effect will be larger than the first effect, so MR will be decreasing as the quantity sold increases.

Calculus approach
The main goal of a firm is to maximize profit, which is given by:
 * $$\pi(Q)=R(Q)-C(Q) = p(Q)Q - C(Q)$$

To find the output which maximizes profit, we take the derivative of the profit function with respect to quantity:
 * $$\frac{\delta\pi}{\delta Q} = \frac{\delta p}{\delta Q}Q + p - \frac{\delta C}{\delta Q}$$

By setting the derivative equal to zero, we get:
 * $$\frac{\delta p}{\delta Q}Q + p = \frac{\delta C}{\delta Q}$$

There's a few things to notice about that condition.
 * 1) The left-hand side is marginal revenue (MR), and the right-hand side is marginal cost (MC).
 * 2) Marginal revenue comes in two parts. $$\frac{\delta p}{\delta Q}$$ will be negative, since the quantity sold can only increase when the price decreases. So $$\frac{\delta p}{\delta Q}Q$$ is the part of marginal revenue where revenue is lost on units which could be sold at a higher price. Then there's just $$p$$, which is the additional revenue from selling a unit which could not be sold at a higher price. Given the downward sloping demand curve marginal revenue is always less than the price.

Elasticity
We will often use elasticity of demand to discuss the shape of a demand curve, and the elasticity will influence a monopolist's profit-maximizing behavior. Intuitively, if demand is price inelastic, then a firm could raise its price and quantity demanded would fall only slightly. This would be moving upward on the firm's demand curve. At the same time, moving upward on the demand curve will usually mean increasing elasticity, so the monopolist cannot keep raising its price this way forever.

If demand is price elastic, then a monopoly could lower its price a little and have the quantity demanded increase a lot. This would mean moving downward on the demand curve and also usually cause elasticity to decrease. Both of these forces push the monopolist toward a "middling" elasticity of demand, but which elasticity is the most profitable will depend on the firm's costs.

Demand elasticity ($$\epsilon$$) is the percentage change in quantity demanded for a percentage change in price:
 * $$\epsilon = \frac{\%\Delta Q}{\%\Delta p}$$

We'll primarily be using the calculus version which means the same thing, but is continuous:
 * $$\epsilon = \frac{\delta Q}{\delta p}\frac{p}{Q}$$

We can take the marginal revenue we found, $$MR = \frac{\delta Q}{\delta P}Q + p$$, and rewrite it in terms of elasticity:
 * $$MR = \frac{\delta p}{\delta Q}Q + p = p\left(\frac{\delta p}{\delta Q}\frac{Q}{p} + \frac{p}{p}\right)$$
 * $$MR = p\left(\frac{\delta p}{\delta Q}\frac{Q}{p} + 1\right)$$
 * $$MR = p\left(\frac{1}{\epsilon} + 1\right)$$

So far, that is just another way to write about the shape of the demand and marginal-revenue curves, and doesn't say how the firm should maximize profit. However, putting that together with the MR=MC condition:
 * $$MR = MC$$
 * $$p\left(\frac{1}{\epsilon} + 1\right) = MC$$

Remember that elasticity is usually negative, so we can see pretty quickly that if MC = 0, the most profitable elasticity is $\epsilon = -1$. This is called "unit elastic", and means demand is neither elastic nor inelastic. This is of particular interest for information goods, because in many cases, we will want to treat marginal cost as zero.

More generally, we can see that if $MC \geq 0$, then $\epsilon \leq -1$, so demand will never be inelastic when a firm is maximizing its profit. Intuitively, we can think that a monopolist might like inelastic demand because it means they can raise their price without losing many sales. However, because they like it and take advantage of it, they will keep raising their price and elasticity will rise until demand is at least unit elastic and possibly elastic.


 * If demand is elastic, marginal revenue is positive ($$\epsilon < -1$$)
 * If demand is unit elastic, marginal revenue is zero ($$\epsilon = -1$$)
 * If demand is inelastic, marginal revenue is negative ($$\epsilon > -1$$)