Endogenizing Sticky Prices in a Relationship Model

One of the main reasons that firms do not raise prices, according to survey data, is that they are worried about disrupting customer relationships. Yet I haven't seen a model that takes this seriously.

A successful model would have customers that buy one (or many goods) from a known retailer at some price p = (1+mu)*MC which is set by the retailer. The customers would have the option of buying from the retailer at price p or investing in a search at some some cost s to find a new retailer. The customer would have to form some expectation about the price other retailers are charging (some sort of information extraction problem).

The customer then is balancing the expected benefit of finding a lower price with the (known) cost of executing the search. The information available to the consumer might be some sort of productivity shock that affects the marginal cost for retailers in an idiosyncratic way.

The retailers, then, would face an idiosyncratic marginal cost (known only to them) and would choose the markup mu every period. The retailers would then balance the marginal benefit (or cost) from raising (or lowering) the price with the cost (or benefit) of losing (or gaining) customers. The intuition for sticky prices should be fairly clear. A small increase in marginal cost won't lead to a change in prices because it would lead to customers to leave and look for a new retailer. A small decrease in marginal cost won't lead to a drop in price because retailers would have nothing to gain.

There are other things that could (should) be considered in this type of model. Should retailers be able to advertise (gaining customers who are paying more) at some convex cost. This would add realism and would lead to price drops as well as price increases (I assume they wouldn't advertise price increases unless it represented a lower price compared to everybody else).

I could assume that customers have an idiosyncratic reservation price (i.e. they don't buy unless the price is less than their reservation price). Customers could only buy one unit so that retailers focus on gaining more customers rather than selling more to their current customers. The distribution of the reservation prices might be known so that retailers can form a conditional expectation of how much quantity would change if they changed their prices from p to p'.