# Price discrimination is efficient when expanding available markets

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## Statement

When a seller practices price discrimination in a manner that brings more buyers into the market, the market becomes efficient, because there are more mutually beneficial market exchanges.

## Examples

### An example with constant costs of production per piece

Suppose a seller produces a good at a price of $x$ money units per piece, and there is no competition. Buyer $A$ is willing to pay a maximum of $y$ units for the good, and buyer $B$ is willing to pay $z$ units for the good. In other words, the reservation prices of buyers $A$ and $B$ are $y$ and $z$ units respectively.

#### Scenario 1: Monopoly pricing out of one buyer without price discrimination

We first consider the case where $\! z > y > x$ and $\! z - x > 2(y - x)$. For instance $\! x = 5, y = 10, z = 30$.

Situation How the seller acts and buyers respond Producer surplus Consumer surplus Social surplus
Single price chosen by seller for all buyers (no price discrimination) Seller prices good at $z$, $B$ buys, $A$ doesn't. $\! z - x$ $\! 0$ $\! z - x$
Separate price for each buyer Seller prices at $y$ for $A$ and $z$ for $B$, both buy. $\! y + z - 2x$ $\! 0$ $\! y + z - 2x$
Perfectly competitive market seller prices at $x$ units, both $A$ and $B$ buy $\! 0$ $\! y + z - 2x$ $\! y + z - 2x$

In this case, without price discrimination, there is only one buyer in the market, and the social surplus is $z - x$, i.e., the surplus when buyer $B$ buys the good. However, with price discrimination, a profit-maximizing seller can sell to both $A$ and $B$, and the social surplus is now $z + y - 2x = (z - x) + (y - x)$.

#### Scenario 2: Profitable to sell to both buyers even without price discrimination

We next consider the case where $\! z > y > x$ and $\! z - x < 2(y - x)$. A numerical example would be $\! x = 5, y = 10, z = 12$.

Situation How the seller acts and buyers respond Producer surplus Consumer surplus Social surplus
Single price chosen by seller for all buyers (no price discrimination) Seller prices good at $y$, $A$ and $B$ buy. $\! 2(y - x)$ $\! z - y$ $\! y + z - 2x$
Separate price for each buyer Seller prices at $y$ for $A$ and $z$ for $B$, both buy. $\! y + z - 2x$ $\! 0$ $\! y + z - 2x$
Perfectly competitive market seller prices at $x$ units, both $A$ and $B$ buy $\! 0$ $\! y + z - 2x$ $\! y + z - 2x$

In Scenario 2, there is no difference in social surplus between the single-price case and the case of a separate price for each buyer. The difference between the two cases within Scenario 2 is that in the single-price case, the buyer with the higher reservation price captures part of the social surplus. Thus, in Scenario 2, price discrimination does not expand markets and hence does not increase efficiency; it simply results in a redistribution.

### An example with changing costs per piece

Suppose a seller produces a good at a price of $x$ money units for the first piece and $w$ money units for the second piece, and there is no competition. Buyer $A$ is willing to pay a maximum of $y$ units for the good, and buyer $B$ is willing to pay $z$ units for the good. In other words, the reservation prices of buyers $A$ and $B$ are $y$ and $z$ units respectively.

#### Scenario 1': Monopoly pricing out of one buyer without price discrimination

If $\! z > y > w, z > x$ and $\! z - x > 2y - (x + w)$, the seller will price the good at $z$ and sell only to $B$, whereas in the price discrimination case, the seller prices at $y$ for $A$ and $z$ for $B$.

Situation How the seller acts and buyers respond Producer surplus Consumer surplus Social surplus
Single price chosen by seller for all buyers (no price discrimination) Seller prices good at $z$, $B$ buys, $A$ doesn't. $\! z - x$ $\! 0$ $\! z - x$
Separate price for each buyer Seller prices at $y$ for $A$ and $z$ for $B$, both buy. $\! y + z - (x + w)$ $\! 0$ $\! y + z - (x + w)$

#### Scenario 2': Profitable to sell to both buyers even without price discrimination

On the other hand, if $\! 2y - (x + w) > z - x > 0$ and $\! z > y > w$, we have:

Situation How the seller acts and buyers respond Producer surplus Consumer surplus Social surplus
Single price chosen by seller for all buyers (no price discrimination) Seller prices good at $y$, both $A$ and $B$ buy $\! 2y - (x + w)$ $\! 0$ $\! 2y - (x + w)$
Separate price for each buyer Seller prices at $y$ for $A$ and $z$ for $B$, both buy. $\! y + z - (x + w)$ $\! 0$ $\! y + z - (x + w)$

We now make an observation: The larger the value of $w$, the more likely that we are in Scenario 1', where price discrimination does expand markets. On the other hand, the smaller the value of $w$, the more likely it is that we are in Scenario 2', where price discrimination does not expand the market. In particular, in the extreme case of $w = 0$, which is realized for some knowledge goods, price discrimination appears to be the least likely to expand markets.

(This needs to be reconciled with the fact that price discrimination is often seen more in knowledge goods. There are many possible reasons for this. First, knowledge goods may have players with more market power in some ways. Second, the difference in people's reservation prices for knowledge goods may be substantially larger than for other kinds of goods. This is accentuated further by the fact that knowledge goods can often be sold across a very wide geographical sweep because knowledge travels cheaply).