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?
Lauwrence Lau (1969) has pointed out that &quot;Samuelson has pointed out the basic duality that exists between the direct and indirect utility functions: They are connected by Legendre’s duaI transformation.&quot;
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.