1. Introduction
The marginal revolution occurred in the 1870s. Political economists tried to extend supply and demand reasoning to both the short run and the long run. They erroneously described the economic problem as allocating given supplies, of labor, land, and capital, among alternate uses. They generalized the principle of substitution. Higher prices of a good are supposed to induce consumers and producers to substitute other goods for the more expensive good.
These confused ideas became accepted partly as a response to Marx. They do not really refute Marx, of course. They merely offer a contrast.
Anyways, the marginal revolutionaries were shown to be mistaken over half a century ago. Many microeconomic textbooks are just wrong.
I like to echo these demonstrations by outlining numerical examples with upward-sloping demand functions for labor. In these examples, managers of firms choose to adopt more labor-intensive techniques, out of an existing menu of choices, at a higher wage. The examples apply under ideal assumptions of competitive markets, no information problems in the long run, no search costs in the long run, and so on. Opocher and Steedman (2015) is an example, out of a vast literature, on which I draw. I like to cite Borjas' textbook for muddled nonsense.
Anyways, I have posted four examples here, for those willing to work through a bit of math.
2. An Example With Natural Resources
In this example, given endowments of land limit how much can be produced. Even with the dependence on final demand, the noted problem with labor economics arises.
3. An Example with Three Produced Commodities
You could check this example with a spreadsheet. Answers are given, so it can be checked with arithmetic. I outline something like a market algorithm when the wage is given. Two levels of the wage are given. (An example of supposedly educated economists being unwilling or unable to recognize results established sometime last century is here.)
4. An Example with Fixed Capital
This post works through a fixed capital example from Baldone (1974), in Studi Economici. A machine that lasts for three production cycles is an example of fixed capital. I take the interest rate as given. To check, you have to be willing to solve a linear system of four equations in four variables. (I do not go into the centre of a fixed capital system, which reduces that system to one of two equations in two variables.) This exposition asserts that firms will not produce commodities with negative prices.
5. An Example With Two Commodities
This exposition works through an example from Bruno, Burmeister & Sheshinski (1966, p. 545) in the Quarterly Journal of Economics. These are mainstream economists, and that journal, associated with Harvard, is one of the most prestigious in the world. This example has two produced commodities. The example requires you to solve two equations in two unknowns, given the interest rate (also known as the rate of accounting profits).
6. Conclusion
Many more examples like these can be produced, including with smooth production functions, rent, and so on. Empirical examples have been illustrated in the literature. I would have liked to have seen more. Innovation in actually existing capitalist economies is probably more about creating new processes and products, not the choice from an existing cookbook of recipes, as presented in production functions and the theory of supply and demand.
I want to note these responses and these responses, in comments from confused, presumably academic economists.
TL;DR: Mainstream economists agreed in the 60s of last century that what they teach is incorrect and illogical.