As I have already shown here and here, the fact that firms set the price does not mean that they determine the prices. Under two-sided competition conditions (which is the most common case in our real world), prices are determined by subjective valuations of marginal pairs. When we see firms setting the price, they are just trying to anticipate what that price will be. That's exactly what great austrians demonstrated as we shall see.
Here is Böhm-Bawerk (1891: 204-205) on a footnote:
"The more experienced both parties are, and the more familiar with the condition of the market, the shorter will be the time spent in "trying the market" by preliminary offers. In an old and well-organised market competitors will save themselves the trouble of making offers that are not meant to be taken, and will make their first offers at least somewhere near that zone within which the market price will finally be fixed. The extreme limit of this curtailment is given in the "fixed prices" of sellers. In this case, trying the market is entirely dispensed with, and sellers undertake at one throw, as it were, to hit the very zone into which the condition of the market will force the price. They must try to hit this zone quite exactly; for if they put the price lower they lose their profit, while if they put it higher the buyers in the market get supplied by other competitors, and the sellers are left with their commodities. Fixed prices, however, are less common in the open market than in shops, where selling is never conducted under the full pressure of competition, and where, consequently, any mistake in the price asked is not so hazardous."And Rothbard (1962: 118-19):
"Aside from the universal fact of the scarcity of all goods, a price that is below the equilibrium price creates an additional shortage of supply for demanders, while a price above equilibrium creates a surplus of goods for sale as compared to demands for purchase. We see that the market process always tends to eliminate such shortages and surpluses and establish a price where demanders can find a supply, and suppliers a demand.
It is important to realize that this process of overbidding of buyers and underbidding of sellers always takes place in the market, even if the surface aspects of the specific case make it appear that only the sellers (or buyers) are setting the price. Thus, a good might be sold in retail shops, with prices simply "quoted" by the individual seller. But the same process of bidding goes on in such a market as in any other. If the sellers set their prices below the equilibrium price, buyers will rush to make their purchases, and the sellers will find that shortages develop, accompanied by queues of buyers eager to purchase goods that are unavailable. Realizing that they could obtain higher prices for their goods, the sellers raise their quoted prices accordingly. On the other hand, if they set their prices above the equilibrium price, surpluses of unsold stocks will appear, and they will have to lower their prices in order to "move" their accumulation of unwanted stocks and to clear the market.
The case where buyers quote prices and therefore appear to set them is similar. If the buyers quote prices below the equilibrium price, they will find that they cannot satisfy all their demands at that price. As a result, they will have to raise their quoted prices. On the other hand, if the buyers set the prices too high, they will find a stampede of sellers with unsalable stocks and will take advantage of the opportunity to lower the price and clear the market. Thus, regardless of the form of the market, the result of the market process is always to tend toward the establishment of the equilibrium price via the mutual bidding of buyers and sellers. It is evident that, if we eliminate the assumption that no preliminary sales were made before the equilibrium price was established, this does not change the results of the analysis."
And Mises also knew it in 1912. As we see, the austrian causal-realist subjective marginalist theory of price determination is far superior to mainstream neo-classical economics (see for example here, here, here and here) and to the superficiality and empirical inexactness of the so called "heterodox" economics.