# What prices do firms impose on perfect substitutes?

Let's say we have a retail firm, which sells various products.

Among these products, the firm also sells five or so products that are for almost all customers perfect substitutes, although the tastes are not precisely known. We can imagine the individual consumer's utility function such as: $$U(\boldsymbol{x}) = a_1 x_1 + a_2 x_2 + \dots + a_n x_n$$, while the firm may have an estimate of $$a_j$$, but for now we could assume the estimate is unreliable.

A) Is it optimal for a firm to set the same price for all of these products?

Then, there is another problem if we reverse the logic. We may come into a shop and we are suspicious that multiple products may be perfect substitutes. Can we infer something about them from their prices?

B) If multiple products have the same price, are these likely to be perfect substitutes with equal tastes?

Regarding point A, my rationale is such that the firm should price them relevant to their tastes. If the firm sets the same price and tastes of one product are significantly above tastes of other products, all consumers would just buy it and the firm could increase the price for this exact product. Similarly, in point B, we would expect that this is an optimal policy that the firm converged to.

However, what if there is a noise? What if the firm knows small part of the population (percentage $$\gamma$$) has tastes $$\boldsymbol{a}^S$$, while the majority ($$1-\gamma$$) has tastes $$\boldsymbol{a}^M$$? Since all products are perfect substitutes, under same prices, the population would pool to buy their relevant products. Shall we expect the firm to set prices accordingly to the relevant shares of tastes in the population, or is it optimal for the firm to just take losses on one product, while supporting it by increased gains of other products, especially, if the taste distribution would be continuous (and not binary as I have delimited here)?