Why Do The Same Products Have Different Prices?

Explaining the mystery of price dispersion

Why are the same goods available at many different prices? It is surprisingly difficult to motivate a model to explain why this should happen – our intuitive go to of “search costs” requires careful treatment to not give weird solutions. In fact, the now default method of modeling monopolistic competition rules out price dispersion, because there are never two goods sold!

I will start with the Dixit-Stiglitz framework, which has become the default. Every firm can costlessly differentiate their good from others, and faces the cost structure A + Bx, where A is a fixed cost to enter. No firm wishes to compete directly with anyone else, so they will produce a different good from everyone else. Consumers have a constant elasticity of substitution between goods; thus, mark-ups are fixed as a percentage of the good’s price. You can represent different industries by nesting the CES functions, and having one elasticity govern switching between firms in an industry and switching between industries. This rules out price dispersion of the same good, because each producer would sell only at their own profit-maximizing price. In fact, it rules out any difference in price whatsoever, with homogenous firms, because every firm faces the same cost-structure and charges the same markup!

The first work in explaining price dispersion was by George Stigler, and argued that it was due to information being a good like any other. Consumers face a known distribution of possible prices, and pay a cost to find information. Consumers maximize their welfare, not how long the price they pay is, and so he predicts that the dispersion of prices paid will be wider when the cost of finding information is higher.

We need to specify how search occurs. If it is the case that the consumer solicits the prices of more than one firm, then a standard result of price competition is that the number of firms doesn’t matter for the price. The two companies will compete each other down to 0 profits. (Obviously, we will need to append some assumptions about how the consumer can out wait the firms, and that the firms do not coordinate). Search frictions don’t matter in that world.

What about if consumers search sequentially? They go to a firm, evaluate whether they are willing to pay that price, and pay a fixed cost to search again. The weird outcome of this is that, as Diamond (1971) showed, the presence of any positive search cost leads to the only equilibrium being the monopoly price! Since everyone knows the process which people will go through, consumers don’t expect to find anything better if they search elsewhere, so nobody searches.

To get around this, some models impose heterogeneity on consumers. In Salop (1977), price dispersion acts as a tax upon searchers who face higher costs of looking. However, I find that kind of boring – of course you can get price dispersion if you just say that consumers aren’t identical! You could do the same thing by saying that firms face heterogenous costs. Instead of having ex ante heterogeneity, we can have ex post heterogeneity in the information received, and dispersion is the result of the only Nash equilibrium being a mixed strategy one.

The two most famous papers on this are Butters (1977) and Burdett and Judd (1983). Butters has the consumer be totally passive, and firms choose to send out advertisements which reach a consumer at some cost. The buyer then buys the lowest price they see. This leads to the optimal strategy for the firm being to randomize its posted prices. Interestingly, high-priced sellers will advertise more than low-priced sellers will.

Burdett and Judd is cleaner, and does not imply different strategies for different numbers of consumers and firms. Instead, there is a parameter alpha which stands in for the cost of searching, and is between 0 and 1. Consumers can choose to send a letter to one or two firms to inquire about prices, and then they buy. Firms don’t know how many offers the person is considering, so it is profitable to undercut. In equilibrium, price dispersion is fully characterized by the search cost – if it is 0, then we have the perfect competition price, and if it is 1, we have the monopoly price. In between, the distribution of prices chosen shifts from one to another. They also show that if there is a noise in search, so that someone who goes out searching for one price accidentally finds two, the natural outcome is price dispersion.

One naturally wonders what the implications of advertising are in oligopolies. As Butters showed, the outcome is in fact efficient with free entry. To see what happens if we restrict entry, we must turn to Grossman and Shapiro (1984). Think of two firms which exist on a plane in the style of Hotelling. You can imagine two hot dog vendors on opposite sides of a circle, which we’ll denote 0 and 1. They have no marginal cost of production, and face transportation costs which are linearly increasing with distance. The firms send out advertisements, and consumers choose to buy based on this.

We get some conclusions out of this. The price is higher than if firms did not need to advertise. If “transport cost” is high (here, it’s not so much a literal transport cost, but how different the goods are from each other) then we will see more advertising. This implies that differentiation causes advertising, not the other way around. If we increase the cost of advertising, then profits actually increase. I think this is for similar reasons as Brander and Krugman (1983). Businesses are incentivized to advertise simply to steal a customer from other businesses, which is socially wasteful, and so high advertising costs could raise total welfare. (Though not consumer welfare!)

These models are quite hard to operationalize. It isn’t clear that alpha in Burdett and Judd can be made meaningful, for example. The models give some predictions which don’t seem easily accurate, such as that the price dispersion of goods should not vary by price. They are, nevertheless, beautiful to me. I like thinking about this. It’s very clean.