Basic Pricing Theory
Revenue Maximization, for a Single Price
For a linear Demand Curve such as that below, it can be shown by Calculus (or equivalently, through an Optimization software program such as Excel Solver) that if a single price must be charged, Revenue is maximized at the midpoint, at point B, for $20 x 20 units = $400. We will show later how this is related to Monopoly Pricing theory.
First, Second, and Third Degree Price Discrimination (To remember which is stronger, just think of "Murder in the First Degree"). The first two are simple:
In 1st degree price discrimination, the producer gets all the surplus -- a different price for every customer. For the above demand curve, and if there were a very large number of customers distribute all along the demand curve, this would mean charging the least price-sensitive customers $39.99, the second least price-sensitive customers $39.98, etc., all the way down to charging the cheapest customers $0.01. Note that this would gather a revenue of approximately ($40 x 40 units)/2 = $800.
In 2nd degree price discrimination, by basing price on the quantity bought (volume discounts), the 1st degree of discrimination (a different price for every customer) is approximated by having several prices for different quantity classes of customers. That is, for the demand curve below, instead of just selling at a single price, at point B ($20 x 20 units = $400) we sell at A, B, and C: the first 10 units at $30 each, the next 10 units at $20 each, and the next 10 for $10 each. It is not difficult to see that this gains total revenue of $600, or $200 extra from the single price maximization.
Combining these two markets, the total demand curve looks like the picture below, with two different prices picked off, at B and B'
PRODUCT BUNDLING
The following practice is increasingly common, given that firms are better able to process information about different demand patterns. Suppose Microsoft estimates the following sorts of broad customer "types" for its several types of 'Office' software, Word, Excel, Power Point, and Access. The dollar amounts are meant to show the most that the average customer of this type is willing to pay for a particular application. Hence, the average HR type is willing to pay $100 for Word, but only $40 for Access, while the average Consult type is willing to pay only $70 for Word and $100 for Access.
Note that if MS wishes to sell to all 4 types, and sell each product singly, the most it can charge is $220 -- the minimum that each type is willing to pay for each product. If they can bundle and sell to all types, however, MS can charge $300, the minimum that any group is willing to pay for the entire bundle. This nets an extra $80 per bundled product. (To get a spreadsheet version of the above table, go to Bundling.xls.)
Note that this is not the same as price discrimination, because all customers are being treated the same. If MS were able to price discriminate, then it would charge $300 to the HR types, $310 to Engineer types, $340 to Sales types, and $330 to consult types. However, what it does have in common with price discrimination, is being able to distinguish different types of customers in the aggregate, though not necessarily as individuals.
Bundling is not, strictly speaking, price discrimination, since the above example does not rely upon knowing what any individual's willingness to pay is. People are known only statistically as types, but cannot be identified individually as one type or another. However, bundling can be used to attain the same ends that price discrimination, without making the price discrimination look so explicit. The following example is taken from a paper by Andrew Olyzko, "Privacy, economics, and price discrimination on the Internet," http://www.dtc.umn.edu/~odlyzko/doc/privacy.economics.pdf :
Consider an example of site licensing, which is really a form of bundling. Suppose Alice has a software package to sell, and a company she would like to sell it to. Of the company's 1000 employees, 900 have no interest in Alice's program, 10 of them are willing (or their bosses are willing) to pay $10 apiece, 10 are willing to pay $20 apiece, and so on at each $10 price break, up to 10 who are willing to pay $100 apiece for the program. If Alice knows these valuations, and has to sell to individuals at a fixed price, the optimal choice for her is to charge either $50 or $60 for her package. In either case she will get $3,000. However, the collective valuation of all the employees in this company is $5,500, so she should be able to sell the package for unlimited use by every one of the 1,000 employees for $5,500. Thus by selling a site license, Alice will actually do as well as if she could charge each individual that person's valuation for her package. At the same time, she will appear to be offering the company a bargain. The package, which might sell to individuals outside for $50 per copy or more, will be available at a cost per eligible employee of just $5.50.
The data described above look like this:
Customer's Price Total Customers Added Customers Var. Price Marginal Revenue Var. Price Cumulative Revenue Fix Price Marginal Revenue Fix Price Cumulative Revenue $110 0 0 $0 $0 $0 $0 $100 10 10 $1,000 $1,000 $1000 $1,000 $90 20 10 $900 $1,900 $800 $1,800 $80 30 10 $800 $2,700 $600 $2,400 $70 40 10 $700 $3,400 $400 $2,800 Fix Price > Rev Max> $60 50 10 $600 $4,000 $200
$3,000 < Fix Price < Rev Max $50 60 10 $500 $4,500 $0
$3,000 $40 70 10 $400 $4,900 -$200 $2,800 $30 80 10 $300 $5,200 -$400 $2,400 $20 90 10 $200 $5,400 -$600 $1,800 $10 100 10 $100 $5,500 -$1000 $1,000 $0 0 0 $0 $0 $0 $0 ^ Variable Price Rev Max
The idea is that a Fixed Price Cumulative (total) Revenue (last column) is just the Price times the total customers at that price -- the entries in the 1st and 2nd columns. We see that this is maximized at $3,000, whether we charge $60 each to 50 customers, or $50 each to 60 customers. If we can charge a different price to each customer, however -- perfect price discrimination -- then the marginal revenue for variable prices (2nd to last column) is given by multiplying the entry in the 3rd times that in the 4th column. So we see that the first 10 customers are willing to pay $100 each for $1000, while the next 10 are willing to pay $90 each for an additional (marginal) $900. Using these variable price figures, we determine that as long as there is some positive revenue, we should keep on lowering the price, until we get to a zero price. Thus we get a perfect price-discrimination revenue of $5,500.
This is what the demand for the product would look like, a simple step-function:
% Markup as an index of Market Power
This just shows what is intuitive, that the percent of markup over marginal cost is a measure of the firms market power. More specifically, the more price inelastic (i.e., price insensitive) demand is, the higher firms will be able to raise their mark up over their marginal cost. The 'rule of thumb' for markups is
Price = Marginal Cost * E/(1+E),
where E is the firm's own price elasticity of demand. (Remember that elasticity, E, is a non-positive number, E </= 0. Generally, the more inelastic the demand (more vertical the demand curve), the lower a negative number E will be, and the higher will be the ratio E/(1+E). For, example, if E = -4, then E/(1+E) = -4/-3 = 1.33, whereas if E = -2, then E/(1+E) = -2/-1 = 2. (This ratio E/(1+E) approaches infinity as E approaches -1.)
To see where this comes from, start with total revenue
TR = P(Q)*Q, where P(Q) is just P = P(Q), or P as a function of Q.
Differentiating by the product rule, we have
MR = P+ ¶ P/¶ Q*Q = P(1+ ¶ P/¶ Q*Q/P) = P(1 + 1/E),
where E = ¶ Q/¶ P*P/Q, or the inverse of the second term inside the parenthesis.
Now we know that a monopolist, or indeed any firm with market power, sets MR = MC.
So this gives
MC = MR = P(1 + 1/E) = P(1+E)/E, or , rearranging things:
P = MC*E/(1+E), where E/(1+E) is the mark-up factor over marginal cost.
