Lecture Notes 16: Price Dispersion
A key assumption any time we talk about competitive markets is that consumers have full information about prices. Consumers know what prices the firms in the market are charging. But practically this assumption can often fail to hold. In particular, it might be costly for consumers to search for information about prices – they have to travel, spend time gathering information about prices, etc… In this unit, we will first consider whether the existence of search costs can lead to prices above the competitive level. Then we will consider a model that can generate different prices in equilibrium, even for exactly the same product.
Tourist-Trap Model
The tourist-trap model is due to Diamond (1971). Consider a market for lemonade in a large city full of tourists. There are many lemonade stands. The market has the following characteristics.
1. Consumers view the lemonade from each stand to be identical.
2. All firms have the same cost associated with selling a cup of lemonade. 3. All consumers have the same demand function.
4. Tourists know the general distribution of prices, but do not know before visiting a stand what price each stand charges.
5. Tourists incur a search cost 𝑐𝑐 for each stand they visit. For example, if a tourist visits 3 stands before buying lemonade, her search costs are 3𝑐𝑐.
We will call the competitive price in this market 𝑃𝑃𝑐𝑐 – the price that would generate zero profits for each firm. We will call the monopoly price 𝑃𝑃𝑚𝑚 – the price that would maximize profits for each firm if each customer was captive.
Let’s start by supposing that each firm charges 𝑃𝑃𝑐𝑐. Is this the equilibrium in the market? No, and here is the problem. Consider a single lemonade stand charging 𝑃𝑃𝑐𝑐. It can increase its price a bit without losing any customers. Specifically, suppose the firm increases its price to 𝑃𝑃𝑐𝑐 +𝜀𝜀, where 𝜀𝜀 <𝑐𝑐. Think about the situation of a customer – even if every other lemonade stand is charging 𝑃𝑃𝑐𝑐, it’s not worth it for our customer to go search for another stand because the cost of continuing
to search is more than the reduction in price. Our customer might be annoyed, but it’s not worth it for her to go search out another lemonade stand.
In summary, it cannot be an equilibrium in this market for all firms to charge the competitive price. This is because each firm on its own would have a profitable deviation to raise price a bit – it loses no customers by doing so, as long as the price increase is less than the search cost.
Is it an equilibrium for all firms to charge a price a bit more than 𝑃𝑃𝑐𝑐? Nope – the same argument would hold. In our example above, even if all lemonade stands were charging $4.50, that’s still not an equilibrium. A single stand could then raise its price a bit above the new price, and the same argument would hold. The stand gets more money from each customer without losing any customers in the process.
So what is the equilibrium? Surprisingly, the only equilibrium in this model is for each firm to charge the profit-maximizing monopoly price. In other words, each firm sets price as if it has each customer totally captive and faces no competition.
For our example, suppose that the monopoly price is $10. In other words, this is the price that the stand would charge to each customer to maximize profit if it faced no competition. Why is this an equilibrium? Certainly, raising price makes no sense since $10 is the price that maximizes profits. What about cutting price to steal some customers from other stands? The stand would have to cut price by at least the search cost to even have a hope of picking up any new customers, and if there are many stands then in fact a much greater price cut would be required in order for customers to find it worth searching over and over again to find our low-price stand.1 In summary, it makes no sense for a single stand to raise its price and it makes no sense for a single stand to cut its price. This is an equilibrium.
In the tourist trap model, the outcome where all firms charge the monopoly price is the only equilibrium. Even though the market features many firms with identical products and consumers who are not brand-loyal, no equilibrium with lower prices is possible. At any such equilibrium, a single firm could increase its price by an amount less than the search cost to increase profits. This is a profitable deviation and so these prices cannot constitute an equilibrium.
Does it help to reduce search costs? Surprisingly, the answer is no, at least in this model. Suppose for our example above that the search cost for consumers is reduced from $0.50 to $0.25. It doesn’t make any difference. Our proposed equilibrium where all stands charge $5.00 still won’t work because a firm can deviate to $5.24 to increase profit. And reducing the search cost to $0.10 doesn’t help us either, because a firm can deviate to $5.09 and increase its profit.
1 If the number of firms is very low, then it may be worth it for customers to search for a low-price firm. In this case,
At the end of the day, if the competitive price is 𝑃𝑃𝑐𝑐, a firm can always do better by reducing its price to 𝑃𝑃𝑐𝑐 +𝜀𝜀. No matter what the search cost is, a deviating firm can always find a price bump that is less than the search cost. Thus, for any positive search cost, the only equilibrium in this model is for all firms to charge the monopoly price.
The search cost has to go all the way to zero in order to make any difference. If the search cost is zero, then the price drops down to the competitive, zero-profit level. The interesting point here is that any nonzero search cost, no matter how low is enough to break the competitive equilibrium and generate an equilibrium where all firms charge the monopoly price. And reducing the search cost makes no difference unless it goes all the way to zero.
Here is a summary of the main result of the tourist-natives model.
• Suppose consumers have to pay a search cost 𝑐𝑐> 0. If the model has an equilibrium, the only equilibrium is one where all firms charge the monopoly price. Reductions in the search cost have no impact on equilibrium price unless 𝑐𝑐 = 0. If 𝑐𝑐 = 0, then the equilibrium reverts to the competitive equilibrium.
Two quick things to add. First, if advertising low prices reduces a consumer’s search cost to zero to find low-price firms, then the result above says that the price drops from the monopoly price to the competitive price. This is definitely an improvement for society. As we said in the unit on advertising, advertising about price has the potential to improve welfare.
Second, entry in this model can actually be bad. We said above that it might be possible that a firm could break the equilibrium where all firms charge the monopoly price by reducing its price to get some new customers. This will only happen if it is easy enough for consumers to search for the new firm, and the price difference justifies these search costs. A reduction in the number of firms could actually make consumer search easier and break the monopoly-price equilibrium.
Tourist-Natives Model
Here are the basic assumptions of the model.
1. There are 𝐿𝐿 total buyers in the market. A proportion 𝛼𝛼 are informed (zero search costs). A proportion 1− 𝛼𝛼 are uninformed (positive search costs).
2. Each consumer buys 1 cup of lemonade. The maximum he is willing to pay is 𝑃𝑃𝑚𝑚.
3. There are 𝑛𝑛 firms with U-shaped average cost curves. The competitive price 𝑃𝑃𝐶𝐶 is at the minimum of the average cost curve.
It turns out that this model can have multiple equilibria, and the equilibrium structure depends on the relative number of informed and uninformed customers.
First, if the number of informed customers is large then there is an equilibrium where all firms charge the competitive price. To see why, if all firms charge the competitive price, then the business is split evenly, and each firm sells 𝐿𝐿
𝑛𝑛 units of output. But if a single firm raises its price
higher than the competitive level, the firm loses all of its informed customers, so its sales fall to (1− 𝛼𝛼)𝐿𝐿𝑛𝑛. Its average costs at this low level of sales may be greater than 𝑃𝑃𝑚𝑚. If so, the firm loses money by deviating. Basically, if there are enough informed customers, each firm loses too many customers by raising price to make it worth doing.
• If the proportion of informed customers is high enough, there is an equilibrium with all firms charging the competitive price.
On the other hand, if the proportion of informed customers is too low, it is not an equilibrium for all firms to charge the competitive price. A firm can deviate. The firm only loses a few sales by doing so, and fetches a much higher price from the remaining uninformed customers. Thus, there is no equilibrium with all firms charging the competitive price.
There is also no equilibrium with all firms charging 𝑃𝑃𝑚𝑚. If all firms charged 𝑃𝑃𝑚𝑚, then a single firm could deviate by charging just a hair less – and it would instantly capture all of the informed customers. This is certainly a profitable deviation, and so there cannot be an equilibrium with all firms charging 𝑃𝑃𝑚𝑚.
So what is the equilibrium? It’s going to have to involve multiple prices. A fraction 𝛽𝛽 of stores will charge the competitive price 𝑃𝑃𝐶𝐶 and sell 𝑞𝑞𝐶𝐶 units of output. The remaining fraction 1− 𝛽𝛽 of stores will charge the monopoly price 𝑃𝑃𝑚𝑚 and sell 𝑞𝑞𝑚𝑚 units of output. We want to study the proportion of high and low-price firms in equilibrium.
1. High-price firms reach 𝐿𝐿(1− 𝛽𝛽) consumers initially.
2. Of these consumers, only the uninformed ones stay. So the total number of sales for high-price firms is (1− 𝛼𝛼)𝐿𝐿(1− 𝛽𝛽).
3. The number of high-price firms is 𝑛𝑛(1− 𝛽𝛽). Assuming that the business is split evenly, the sales of each high-price firm are therefore:
𝑞𝑞𝑚𝑚 =(1− 𝛼𝛼)𝐿𝐿(1− 𝛽𝛽)
𝑛𝑛(1− 𝛽𝛽) =
(1− 𝛼𝛼)𝐿𝐿 𝑛𝑛
Now let’s think about the same problem for low-price firms.
1. Low-price firms capture all of the 𝛼𝛼𝐿𝐿 informed customers.
2. A fraction 𝛽𝛽 of the (1− 𝛼𝛼)𝐿𝐿 uninformed customers are lucky enough to find a low-price firm initially. The total number of these customers is, therefore (1− 𝛼𝛼)𝐿𝐿𝛽𝛽.
3. The number of low-price firms is 𝑛𝑛𝛽𝛽. Assuming that the business is split evenly, the sales of each low-price firm are therefore:
𝑞𝑞𝐶𝐶 = 𝛼𝛼𝐿𝐿+ (1− 𝛼𝛼)𝐿𝐿𝛽𝛽
𝑛𝑛𝛽𝛽
Note that the low-price firms capture all of the informed consumers and also some of the uninformed consumers who got lucky. Thus, their market share is 𝛽𝛽> 𝛼𝛼.
Now here’s where economics comes in. In equilibrium, both firms have to make zero profit. The low-price firms make zero profit by definition, but then the high-price firms have to make zero profit as well, otherwise low-price firms would do better to deviate and become high-price firms.
Because average cost is u-shaped, there will be some output so low that average costs are equal to 𝑃𝑃𝑚𝑚. It is at this output that the high-price firms make zero-profit. Call this output 𝑞𝑞𝑧𝑧𝑚𝑚.
Using our expression for the output of a high price firm:
(1− 𝛼𝛼)𝐿𝐿
𝑛𝑛 =𝑞𝑞𝑧𝑧𝑚𝑚
Similarly, there will be a (higher) level of output at which average costs are equal to 𝑃𝑃𝐶𝐶. Call this output 𝑞𝑞𝑧𝑧𝐶𝐶. This must be the output produced by each competitive firm to get it to zero profit.
𝛼𝛼𝐿𝐿+ (1− 𝛼𝛼)𝐿𝐿𝛽𝛽 𝑛𝑛𝛽𝛽 =𝑞𝑞𝑧𝑧𝐶𝐶
You can solve the two preceding expressions to characterize the zero-profit market equilibrium total number of firms 𝑛𝑛 and the share of low-price firms 𝛽𝛽. From the first expression:
𝑛𝑛 =(1𝑞𝑞− 𝛼𝛼)𝐿𝐿𝑧𝑧𝑚𝑚
Substituting this back into the second expression gives the equilibrium level of 𝛽𝛽:
𝛼𝛼𝐿𝐿+ (1− 𝛼𝛼)𝐿𝐿𝛽𝛽 �(1𝑞𝑞− 𝛼𝛼)𝐿𝐿𝑧𝑧𝑚𝑚 � 𝛽𝛽
= 𝑞𝑞𝑧𝑧𝐶𝐶
⇒ 𝛽𝛽= (1− 𝛼𝛼)(𝑞𝑞𝛼𝛼𝑞𝑞𝑧𝑧𝑚𝑚𝑧𝑧𝑐𝑐− 𝑞𝑞𝑧𝑧𝑚𝑚)
One important point to emphasize again. Even though the equilibrium involves firms charging two different prices, both firms are at zero-profit. The high-price firm suffers a reduction in output which drives up its average costs. The market equilibrates the number of firms and the proportion of high and low price firms until sales of each type of firm are such that both reach zero profit.
This is a really interesting model for economists. The so-called “law of one price” is usually regarded to be pretty rock solid. The same product should not sell for different prices at different locations. The law of one price has held up quite well empirically in a lot of settings and is the cornerstone that underlies models as diverse as asset pricing and exchange-rate determination like purchasing power parity and uncovered interest parity.
