At Marginal Revolution, E. Glen Weyl tackles the meaning of price theory.
This process eventually brought me to my own definition of price theory as analysis that reduces rich (e.g. high-dimensional heterogeneity, many individuals) and often incompletely specified models into ‘prices’ sufficient to characterize approximate solutions to simple (e.g. one-dimensional policy) allocative problems. This approach contrasts both with work that tries to completely solve simple models (e.g. game theory) and empirical work that takes measurement of facts as prior to theory.
Thus, price theory as practiced:
To illustrate my definition I highlight four distinctive characteristics of price theory that follow from this basic philosophy. First, diagrams in price theory are usually used to illustrate simple solutions to rich models, such as the supply and demand diagram, rather than primitives such as indifference curves or statistical relationships. Second, problem sets in price theory tend to ask students to address some allocative or policy question in a loosely-defined model (does the minimum wage always raise employment under monopsony?), rather than solving out completely a simple model or investigating data. Third, measurement in price theory focuses on simple statistics sufficient to answer allocative questions of interest rather than estimating a complete structural model or building inductively from data. Raj Chetty has described these metrics, often prices or elasticities of some sort, as “sufficient statistics”. Finally, price theory tends to have close connections to thermodynamics and sociology, fields that seek simple summaries of complex systems, rather than more deductive (mathematics), individual-focused (psychology) or inductive (clinical epidemiology and history) fields.

I trace the history of price theory from the early nineteenth to the late twentieth when price theory became segregated at Chicago and against the dominant currents in the rest of the profession. For a quarter century following 1980, most of the profession either focused on more complete and fully-solved models (game theory, general equilibrium theory, mechanism design, etc.) or on causal identification. Price theory therefore survived almost exclusively at Chicago, which prided itself on its distinctive approach, even as the rest of the profession migrated away from it.

This situation could not last, however, because price theory is powerfully complementary with the other traditions.
I don't know that price theory, even in the Chicago tradition (which is more about tight priors and rigorous tracing out of the implications of priors than about descriptive analysis, or applied mathematics) became segregated, or simply donned new clothing.  For instance, the University of Wisconsin renamed its Intermediate Price Theory course as Intermediate Microeconomics, but there weren't major changes in the catalog description.  And George Stigler himself, in the preface to the fourth edition of The Theory of Price, (I'd better take care of my copy, there's meaningful resale value in it) writes, "Microeconomics is the mature, stable corpus of economic theory, and its continuity is a reflection of its innumerable and infinitely varied successful applications."

It doesn't matter what you call it.  Exchange, arbitrage, indifference, opportunity cost.  The economist who grasps these things better will write better and teach better.

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