6.2 Price Discrimination
Up until now, when dealing with firms with market power, we’ve assumed that the firm sells one product at a specific price to all consumers. However, this is very much not the case: many firms offer a range of options at different price points, ranging from low-priced “budget” products to higher-priced “premium” versions. Think about any product line you like, from iPhones to airplane tickets to cars: most firms sell a range of options to people at different price points. Why do they do this?
The easy answer is that they want to maximize their profits, and they know that different people have different willingness to pay for quality. However, a difficulty may arise if the firm cannot perfectly observe each consumer’s willingness to pay.
Broadly speaking, we might consider two types of price discrimination:
- First-degree price discrimination: The firm can perfectly observe each consumer’s willingness to pay. Even better, it can charge each consumer a different amount. Its problem is therefore to figure out how to make the most profit from each consumer with perfect information.
- Second-degree price discrimination: The firm cannot observe each consumer’s willingness to pay. Therefore, any type of consumer can buy any good in the product line. The firm’s problem is to determine a product line and price points that create a separating equilibrium in which consumers with a low willingness to pay choose a low-price, low-quality offering, while consumers with higher willingness to pay opt for a high-price, high quality option.
There’s actually a third type, called third-degree price discrimination, in which the firm cannot observe each consumer’s willingness to pay, but they can observe something correlated with that willingness to pay. For example, it can observe that the consumer is a student, or a senior citizen; and it can offer discounts based on that factor. But for the purposes of this chapter, we’re going to focus on the distinction between the first two.
Simplest model setup: two types of consumers
A firm is making quality choice with regard to a product line: it can produce goods at various qualities. We’ll denote the quality of product $i$ as $q_i$.
There are two kinds of customers, who differ according to their valuation of this type of product. Specifically, there are $n_1$ type-1 consumers value quality according to the “total benefit” payoff function \(TB_1(q) = 20q - \tfrac{1}{2}q^2\) and there are $n_2$ type-2 consumers value quality according to the payoff function \(TB_2(q) = 30q - \tfrac{1}{2}q^2\) Note that type-2 consumers have a higher willingness to pay than type-1 consumers.
The firm will choose a product line made of two offerings: a “budget” product of low quality $q_1$ and low price $p_1$, aimed at type-1 consumers; and a “premium” product of high quality $q_2$ and high price $p_2$, aimed at type-2 consumers.
For simplicity, we’ll assume the firm has zero costs, so it’s just trying to maximize the revenue from its sales. We’ll write $R$ as the firm’s revenue, with $R_1$ being the revenue earned from type-1 consumers and $R_2$ being the revenue earned from type-2 consumers. Therefore, if the firm is successful in getting low-value types to buy the value option, and high-value type to buy the premium product, then its total revenue will be \(R(p_1,p_2) = \underbrace{n_1p_1}_{R_1} + \underbrace{n_2p_2}_{R_2}\) Again, to keep things as simple for now, we’ll assume there’s just one of each type; so total revenue is just $p_1 + p_2$. (However, some of the homework exercises you’ll see relax this assumption!)
Now that we’ve established the setup of the model, Let’s start out with first-degree price discrimination.
First-Degree Price Discrimination
If the firm can perfectly price discriminate, it can capture all of the surplus in the market: specifically, it can choose a “bespoke” quality level for each type of consumer.
In this case, the firm could charge the entire valuation of the consumer: if it sold a type-1 consumer a good of quality $q_1$, it could charge \(p_1(q_1) = TB_1(q_1) = 20q_1 - \tfrac{1}{2}q_1^2\) Likewise, if it sold a type-2 consumer a good of quality $q_2$, it could charge \(p_2(q_2) = TB_2(q_2) = 30q_2 - \tfrac{1}{2}q_2^2\) One way of thinking about this is to plot each type of consumer’s marginal benefit curve. In this case the total benefit (or gross consumer’s surplus) is the area below the MB curve, up to the quality $q$. For example, if $q = 20$, the total benefit to a person in group 2 would be the area under the MB curve and to the left of $q = 20$, which has an area of $30 \times 20 - \frac{1}{2}20^2 = 400$:
In this case, because the firm can perfectly observe the consumer’s valuation of quality, the firm’s marginal revenue from increasing quality will just be the consumer’s marginal benefit. The firm therefore maximizes its revenue by setting $MR = 0$ for each group: \(\begin{aligned} MR_1(q_1) &= 20 - q_1 = 0 \Rightarrow q_1^\star = 20\\ MR_2(q_2) &= 30 - q_2 = 0 \Rightarrow q_2^\star = 30 \end{aligned}\)
Type-1 consumers value the budget offering with quality $q_1 = 20$ at \(TB_1(20) = 20 \times 20 - \tfrac{1}{2} \times 20^2 = 200\) Likewise, type-2 consumers value the premium offering with quality $q_2 = 30$ at \(TB_2(30) = 30 \times 30 - \tfrac{1}{2} \times 30^2 = 450\) Therefore, the firm’s optimal product line is:
- It sells a budget option with quality $q_1 = 20$ for price $p_1 = 200$.
- It sells a premium option with quality $q_2 = 30$ for price $p_2 = 450$.
Both types of consumers are just barely willing to buy their respective product; and as usual, we’ll assume that if someone is just barely willing to do something, it’s safe to assume they do. (If you prefer, you can imagine that $p_1 = 199.99$ and $p_2 = 249.99$, so that consumers get a penny of surplus.) So if there’s one of each type of consumer, the firm’s total revenue is \(R = 200 + 450 = 650\) Now let’s think about the more interesting case, in which the firm cannot observe each consumer’s type.
Second-Degree Price Discrimination
Key to the firm’s profit maximization in the first-degree price discrimination case was the fact that it could offer each consumer a single product; and as long as it was worth it to buy that product, the consumer would.
However, usually you can’t do that. Apple may offer an iPhone Pro and a normal iPhone, but it can’t prevent someone with a high valuation from choosing the budget option, or vice versa. And in the above example, the type-2 consumers would choose to do just that!
In order to prevent this from happening, we first need to understand the incentives facing the high-valuations customers. Let’s define the consumer surplus from buying a good of quality $q$ for price $p$ as \(S(q,p) = TB(q) - p\) The total benefit from the budget option for a type-2 consumer would therefore be \(S_2(q_1,p_1) = TB_2(q_1) - p_1\) In the above example, with $q_1 = 20$ and $p_1 = 200$, this becomes \(\begin{aligned} TB_2(q_1) - p_1 &= (30 \times 20 - \tfrac{1}{2} \times 20^2) - 200\\ &= 200\end{aligned}\) So a type-2 consumer would receive €200 of surplus from buying the budget option!
That was for a specific case of $q_1 = 20$ and $p_1 = 200$. Let’s assume that the firm charges the type-1 consumers their total willingness to pay. Thus if its budget offering has quality $q_1$, it will set \(p_1(q_1) = TB_1(q_1)\) Plugging this into the first equation, we can write the surplus of a type-2 consumer purely in terms of the quality of the budget option: \(S_2(q_1) = TB_2(q_1) - p(q_1) = TB_2(q_1) - TB_1(q_1)\) Visually, we see this in the following graph. For any quality of the budget option $q_1$, the firm will charge $p_1 = TB_1(q_1)$, which is the area of the red triangle. The surplus to type-2 consumers is the area under its MB curve, up to $q_1$. Therefore the surplus $S_2(q_1)$ is the area below the MB curve of the high types, and above the MB curve of the low types:
So, how does this tie back into the firm’s pricing problem? In order to get type 2’s to buy the premium product, it needs to make them derive at least as much surplus from that product as they do from the budget option: \(\begin{aligned}S_2(q_2,p_2) &\ge S_2(q_1,p_1)\\ TB_2(q_2) - p_2 &\ge S_2(q_1,p_1)\\ p_2 &\le TB_2(q_2) - S_2(q_1,p_1) \end{aligned}\) In simple terms, the firm has to reduce the price of the premium product by the amount the high-valuation buyers could get in surplus by buying the budget option.
So, how does it do this?
By making the budget option lower quality.
Think about it: why are coach seats so uncomfortable? It’s to make them a worse alternative to more expensive seats for those who would consider shelling out the big bucks for first class or even premium economy. The worse the quality of the budget option, the more demand there is for the premium product. And conversely, the nicer the budget option, the less the firm can pay for the premium.
Once we have this insight, we realize that we can write the firm’s problem entirely in terms of $q_1$. For any $q_1$, we know that \(p_1(q_1) = TB_1(q_1)\) Plugging this into the condition \(p_2 \le TB_2(q_2) - S_2(q_1,p_1)\) and recognizing that this will be met with equality, we have \(p_2(q_1) = TB_2(q_2) - TB_2(q_1) - TB_1(q_1)\) Therefore the total revenue for the firm is the sum of the shaded areas in the graph below. Move the dot left and right to find the value of $q_1$ that maximizes the firm’s profits:
OK, so now let’s do the math. Noting that $q_2$ will be constant at 30 (because there are no costs, there’s no reason not to keep the premium product high quality), we have \(\begin{aligned} TB_1(q_1) = 20q_1 - \tfrac{1}{2}q_1^2\\ TB_2(q_2) = 450 TB_2(q_1) = 30q_1 - \tfrac{1}{2}q_1^2\\ \end{aligned}\) Since $p_1 = TB_1(q_1)$, the surplus type-2 buyers would get from buying the budget offering would be \(S_2(q_1) = TB_2(q_1) - p_1 = [30q_1 - \tfrac{1}{2}q_1^2] - [20q_1 - \tfrac{1}{2}q_1^2] = 10q_1\) Visually, this is the area of the gray “discount” indicated in the diagram above. Therefore \(p_2(q_1) = 450 - S_2(q_1) = 450 - 10q_1\) which is the entire blue triangle minus the discount.
Assuming there is one consumer of each type, the firm’s revenue is therefore \(\begin{aligned} R(q_1) &= p_1(q_1) + p_2(q_1)\\ &= (20q_1 - \tfrac{1}{2}q_1^2) + (450 - 10q_1)\\ &= 450 + 10q_1 - \tfrac{1}{2}q_1^2\end{aligned}\) Taking the derivative of this with respect to $q_1$ and setting it equal to zero gives us \(\begin{aligned} R'(q_1) = 10 - q_1 &= 0\\ q_1^\star &= 10\end{aligned}\) And sure enough, if you set $q_1 = 10$ in the graph above, you can see that the total revenue is maximized at 500.
Intuition: Optimal Crappiness
The key intuition here is that the firm faces a constraint: in order to get the type-2 consumers to buy the premium product, it has to offer them a discount off their valuation equal to the amount they would enjoy the budget option. So, lowering the quality of the budget option allows them to charge more for their premium offering. However, lowering the quality of the budget option does reduce the amount they can charge the type-1 consumers for it; so as usual, we face a balancing act. The optimal quality is found when the revenue lost from the lower price for the budget offering exactly offsets the revenue gained by being able to charge more for the premium product.