## Tuesday, November 19, 2013

### In which I think about monopolies

This is part 2 of my previous post.

Why, in a competitive market, are the quantity and price at the intersection of supply and demand?  The answer is that buyers and sellers are not monolithic entities.  As long as the price is above the intersection, a new seller can make money by producing new goods at the marginal cost and selling at the marginal value, even though this lowers the price, resulting in net losses for other sellers.  As long as the price is below the intersection, a new buyer can produce value by paying more for goods, even though this makes it more expensive for all other buyers.

On the other hand, if sellers really are a monolithic entity, then we have a monopolistic market rather than a competitive one.  Under monopoly conditions, the single seller may not wish to sell as much, since whenever the prices decreases, they eat all the losses themselves.

The single seller is free to set the price such that their utility US(p,q) is maximized.  The constraint is that they cannot sell more than buyers demand.  US is defined for all p and q, but we want to determine the maximum US specifically along the demand curve (which as I explained previously, is the same as the marginal value curve).1 $\frac{\mathrm{d} \text{US}}{\mathrm{d} q}|_{p=\text{MV}(q)} = 0$ $\frac{\mathrm{d} \text{US}}{\mathrm{d} q}|_{p=\text{MV}(q)} = \frac{\mathrm{d}}{\mathrm{d} q}(\text{MV}(q)*q) - \text{MC}(q)$ $\text{MV}(q)*q$ is the seller's revenue, the price of goods times the total quantity sold.  $\frac{\mathrm{d}}{\mathrm{d} q}(\text{MV}(q)*q)$ is therefore the marginal revenue.  So in a monopoly, the quantity of goods is at the intersection of marginal revenue and marginal cost.  The price of goods is set to the maximum that demand will allow.  Let's look again at figure 1.

Figure 1: Monopoly

In a competitive market, the quantity and price are given by the intersection of supply and demand, or marginal cost (MC) and demand (D).  In a monopoly market, the quantity is given by the intersection of marginal cost (MC) and marginal revenue (MR), and the price is given by the demand (D) curve.

Figure 2: Monopsony

I've left out analysis of a monopsony, because it's exactly the same.  Instead of a single seller, there is a single buyer who chooses the price to maximize UB while constrained to the supply curve.

In the previous post, I said it was confusing that the "marginal cost" in figure 2 means something completely different from the "marginal cost" in figure 1.  In figure 1, the marginal cost is the cost to the seller to produce one more unit.  In figure 2, the marginal cost is the cost to a single buyer to buy one more unit (including the additional cost from prices being raised).

Next I want to discuss "deadweight loss", which is the sum total utility (of both the buyers and sellers) lost compared to a competitive market.  When people trade goods for money, both the buyers and sellers benefit.  For the buyer, the goods have more value than the money paid.  For the seller, they are paid more than it cost to produce the good.  I already showed this in my previous post when I gave expressions for utility functions UB and US.  However, it helps to have a graphical interpretation of the utility.

Figure 3: Visual representation of utility to buyers (a) and sellers (b)

First, consider the buyer (figure 3a).  The total value of the goods to the buyer is given by the area under the marginal value curve (red).  But to get the total utility we must also subtract off the price paid for the goods (blue).  Next consider the seller (figure 3b).  The cost of production is given by the area under the marginal cost curve (green).  The utility to the seller is equal to the total price paid to buy the goods (blue) minus the total cost to produce them (green).

In figure 1, the utility to buyers is shown in red (consumer surplus), while the utility to sellers is shown in blue (producer surplus).  Relative to a competitive market, a monopoly has two effects.  First, the producer gets more utility while the consumer gets less.  Second, the sum utility is smaller, by an amount shown by the yellow "deadweight loss" area.

Now that I've learned about the concept of deadweight loss, I understand a bit why some people think a "free market" is best (although clearly there need to be restrictions on monopolies, so I think "free" is a misnomer).  But given that this is only the simplest market imaginable, it's not clear that it generalizes to more realistic markets.

Even in this simplest case, I question the meaningfulness of the "total utility".  I think, for instance, there is some intrinsic utility to a more fair or more even distribution of wealth.  The distribution of wealth that results from a competitive market basically bears no relation to what is fair.

And there's another problem that comes from the initial uneven distribution of wealth.  The utility to the consumers is not measured directly, but implied by how much money consumers are willing to spend.  But if you think about why some consumers are less willing to pay for a good, it's not necessarily because they derive less value from the good.  Often it's because they are poorer, and therefore place a higher utility value to the dollar.  Thus the utility to the buyers is not equal to the amount of money they would pay.  It only seems that way if you convert utility to dollar units, ignoring that dollar units are larger to some people than to others.

One way you can fix the problem is by redistributing all wealth evenly.  I think there might be a few kinks in that particular solution though...

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1. This illustrates a mathematical distinction between derivatives and partial derivatives.  When we have a function of two variables, like US(p,q), the derivative isn't uniquely defined unless we specify a particular direction in the (p,q) plane.  So we restrict the (p,q) plane to a particular curve, and find the slope along that curve.  The partial derivative is the slope along a p=constant curve.  But here I take the derivative along the demand curve.

#### 1 comment:

Larry Hamelin said...

Pretty much straight from Principles of Microeconomics.

At the undergraduate level (and, I suspect at even the highest levels), economics does not really strive for accurate prediction; a lot of concepts, such as the utility curve, total utility, and producer and consumer surpluses, assume too many "spherical cows of uniform density" to have quantitative meaning. The graphs (and calculations) are there as intuitive metaphors.

For example, we can't really quantify a deadweight loss, but no matter what its magnitude, eliminating it usually creates a Pareto improvement.

Economists are much humbler inside the profession than we are outside. We admit to ourselves (and the students) that we know a lot less than we say we know in public.