In Reinventing the Bazaar, John McMillan discusses how search costs affect the price the buyer pays. John writes:
“Imagine that all the merchants are quoting $1. Could one of them do better by undercutting this price? There is a downside to price-cutting: a reduction in revenue from any customers who would have bought from this merchant even at the higher price. If information were freely available, the price-cutter would get a compensating boost in sales as additional customers flocked in. When search costs exist, however, such extra sales may be negligible. If you incur a search cost of 10 cents or more for each merchant you sample, and there are fifty sellers offering the urn, then even if you know there is someone out there who is willing to sell it at cost, so you would save $5, it does not pay you to look for him. You would be looking for a needle in a haystack. If you visited one more seller, you would have a chance of one in fifty of that seller being the price-cutter, so the return on average from that extra price quote would be 10 cents (or $5 multiplied by 1/50), which is the same as your cost of getting one more quote. It does not pay to search.”Reinventing the Bazaar, John McMillan
John got it wrong. It pays to search. The cost and the expected payoff for the first quote is 10 cents. But if the first quote is $15, the expected payoff for the second quote—(1/49)*$50—is greater than 10 cents. And so on.
Another way to solve for it is to come up with the expected number of quotes you need to get to get to the seller selling at $10. It is 25. Given you need to spend on average $2.50 to get a benefit of $2.50, you will gladly search.
Yet another way to think is that the worst case is that you make no money—when the $10 seller is the last one you get a quote from. But in every other case, you make money.
For the equilibrium price, you need to make assumptions. But if the buyer knows that there is a price cutter, they will all buy from him. This means that the price cutter will be the only seller remaining.
There are two related fun points. First, one of the reasons markets are competitive on price when true search costs are high is likely because people price their time remarkably low. Second, when people spend a bunch of time looking for the cheapest deal, you incentivize all the sellers selling at a high rate to lower their rates and make it better for everyone else.