It is a bad custom to identify economics by writers' political thought. If economics is a science (as I have used it last time), it should be classified by its theoretical framework. In my understanding, John Hicks has given a good characterization and distinction between classical and neoclassical economics. Classical economics is an economics of production (Plutology) and neoclassical economics is an economics of exchange (Catallactics). If I paraphrase this opposition more concretely, classical economics adopted cost-of-production theory of value and neoclassical economics adopted demand and supply equilibrium as the value determination mechanism.
In my previous post, I explained that the turning point from classical to neoclassical economics can be seen in John Stuart Mill's writings. But I have forgotten to explain how it occurred. John Stuart Mill wanted to solve a problem left unsolved by David Ricardo. There were four of them, but the most important one was the question how the "terms of trade" were determined in international trade. Mill followed Ricardo and considered two country, two commodity case. In searching a situation where two countries gain from trade. In this "classical" (in the sense that this is the most often used setting), Mill was lead to consider a situation where two countries specializes completely. In this situation, the total production volume of each product is determined and Mill was obliged to consider a pure exchange economy. The logic used there was named (later) as reciprocal demand theory. Marshall was much influenced by this construction and he started to consider with demand and supply curves. (For more details, please see my paper.)
The classical and neoclassical economics have a totally different value theory. This determines the basic understanding on how market economy works. Neoclassical economics thinks that market economy is regulated by price adjustment. Classical economists in the 19th century had a double mind. In the time of classical political economy, short period adjustment (less than a year) was done by price fluctuation but Ricardo (and Marx) thought it is better to think that economy is regulated in a long period by values (cost-of-production including profit). Now most of industrial production is adjusted much more rapidly than the 19th century and most of industrial products are now sold at a fixed price (the normal price determined by the full cost). In present-day economy, it is no more price fluctuations which bring demand and supply nearly equal. It is not the market that adjust demand and supply equal but firms which adjust their supply to the demand of the market.
Of course, price does not lose its function. Price system plays as an important guide in the selection of different production processes (or production techniques). This is the very basis how modern economy develops rapidly with technical progress. As the everyday adjustment is done quantitatively, unemployment becomes inevitable when effective demand is not sufficient.
On one of two alternative theoretical cores, one can add many discourses. Standing on the essentially same theory core, some may denounce capitalism and some others defend it as a least bad option.
Professor Emeritus, Osaka City University
Former president of the Japan Association for Evolutionary Economics