# Are Price and Quantity Conjugate Variables?

I am taking a thermodynamics class. In this class one of the things that is discussed is how the partial derivatives of the internal energy with respect to extensive parameters (entropy, volume, mol number, etc) give rise to intensive parameters (temperature, pressure, electrochemical potential, etc). For example for pressure and volume:

$$P = \left(\frac{\partial U}{\partial V}\right)_{S,N}$$

The similarity between pressure volume diagrams and price quantity graphs (see below) got me thinking, is there a similar conjugate relationship between price and quantity? Maybe something like this:

$$P = \left(\frac{\partial U}{\partial Q}\right)_{D^*}$$

Where $$P$$ is the price, $$Q$$ is the quantity, $$D^*$$ is either demand, supply or the conjugate of either and $$U$$ is some sort of economic (as opposed to thermodynamic) potential. This $$U$$ could be found with an intrgral but does it have a name? Can we use Legendre transforms to generate more potentials?  • Are you familiar with supply and demand curves? Oct 16 '20 at 20:33
• Maybe this should be migrated to Economics Stack Exchange? Oct 16 '20 at 21:33
• @Daneel Olivaw Migration sounds good.
– alessandro
Oct 16 '20 at 22:38
• mathoverflow.net/questions/207820/…
– rubikscube09
Oct 17 '20 at 1:20
• @rubikscube09 that is pretty much exactly what I'm thinking here I think.
– alessandro
Oct 17 '20 at 1:34

Lauwrence Lau (1969) has pointed out that "Samuelson has pointed out the basic duality that exists between the direct and indirect utility functions: They are connected by Legendre’s duaI transformation." Lau (1973, chapter 1.3, section 1.4) makes intensive use of the Legendre transformation for deriving dual relationships between production and profit functions. This chapter is available here. Most duality theory between direct and indirect utility functions, and between production, cost and profit functions rely on these relationships (See also Blackorby, Primont and Russell, 1978).

Blackorby, Charles, Daniel Primont, R. Robert Russell, 1978, Duality, Separability, and Functional Structure: Theory and Economic Applications, North-Holland.
Lau, L., 1973, Applications of Profit Functions, in Production Economics: A Dual Approach to Theory and Applications, edited by Melvyn Fuss and Daniel L. McFadden, North-Holland.
Lau, L., 1969, Duality and the structure of utility functions, Journal of Economic Theory, 1, 374-396.

• I just realize that I did not fully answered your question: $U$ should be the utility function, and $P$ the price time the Lagrange multiplier. Oct 17 '20 at 20:39
• Convex duality---e.g. between a convex function and its Legendre transform, or, in the quasi-concave setting, between indirect utility and expenditure functions---is not the type of duality that's being referred to in the question. Oct 18 '20 at 0:20
• Yes, my answer should be improved to show that indeed, price and quantity are Legendre conjugate variables, but this cannot be done without reference to production and profit functions (cost and utility functions). Oct 18 '20 at 10:52

Let p denote pressure and V denote volume then U = p x V where the units of pressure times the units of volume must equal the units for work, heat, and energy.

Tables 2-4 of the NIST Guide to the SI Units show that pressure times volume are equivalent to work, power, and heat expressed as base SI units:

https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes

In physics internal energy U of a system may increase, remain constant, or decrease as the outcome of a natural process however the total of all forms of energy in the system and surroundings must remain constant. This defines a set of natural law relations between the system and surroundings during a process.

Let P denote price and Q denote quantity then Ue = P x Q where the units of price times the units of quantity (edit: revenue) must then be the units for "economic potential".

I would argue that even if price and quantity can be treated like conjugate variables in a model there is still the challenge of establishing a revenue model as an analog to thermodynamic internal energy models within the economic measurement context.

• No curves of price and quantity are not called utility functions. $u(x)= \sqrt(x)$ is a valid utility function even though there is no price. Also $U=PQ$ would not be utility but revenue. Indirect utility is also function of price, but profit function, demand and supply are also functions of price and quantity and they are obviously not utility functions. Oct 18 '20 at 11:04
• @csilvia I will remove my assumption that U is a utility which was based on the comment by Betrand where he states "U should be the utility function, and P the price time the Lagrange multiplier." Oct 18 '20 at 15:51