A securities market, such as for common stocks, comes closest to meeting the criteria for perfect competition. There are numerous participants. Public information is widely available. What is traded is homogeneous. The trade is impersonal and and momentary. Neither trader has any control over the character of the stock. Neither trader has any control over the aggregate supply of, nor demand for, the stock. However, other markets are much different in every way mentioned. Obviously a model of a securities market does not reduce the workings of a market economy as a whole to its essentials.
Austrian economists have pointed out the flaws of the perfect competition model. For example: "Contrary to the perfect competition model, what gives rise to a greater competitive environment is not a large number of participants in a particular market but rather a large variety of competitive products" (link). Also see here with its entertaining narrative about shopping for turkey.
Austrian economist Ludwig von Mises’ fictitious evenly rotating economy is like the stationary state economy described in Perfect Competition #1. An equilibrium of unchanging prices and products and repeating volumes of production and consumption each time period are essential features. Mises says the unchanging evenly rotating economy leaves no room for the entrepreneurial function. See more here.
Economists often write as if an equilibrium is expressible as a set of linear equations assumed to have a solution. For example, see here. However:
1. Getting an exact solution for a set of linear equations in mathematics requires that the independent (input) variables be truly independent.
2. Producers in a market economy are very inter-dependent via supply channels and as competitors.
Accordingly, that seems to turn any solution to a set of linear equations alleged to depict a market economy, or even a small part of one, to be little more than wishful thinking. There are trial-and-error methods of finding a best fit solution to a set of linear equations. Genetic algorithms are probably the main kind. However, the degree of a solution’s fit may be very poor.
P.S. I recently saw a Wall Street Journal article that suggested inter-dependency (link with paywall). The price of Apple stock fell 5% one day after one of its suppliers, Lumentum, cut its earnings and revenue outlook. The article doesn’t explain why. I presumed that many investors assumed Lumentum’s act signaled a cut in Apple’s earnings and revenue outlook, and hence the price drop. Related stories found after seeing the WSJ article: Bloomberg, MotleyFool.