Groucho Marxism

The principle of stationary action is a great unifying principle of physics, and is usually presented in a continuous-time framework. In this blog post I will attempt to present a discrete-time formulation of the principle in accordance with the materialist conception of physics I put forward in a previous blog post. Let x(t) denote the state of a physical system at (discrete) time t. This state may be thought of as a vector x = (x₁,…,x_m), where x_j(t) denotes the position of the particle j in the system at time t. The discrete velocity of the particle j at time t is given by u_j(t) = x_j(t+1)-x_j(t), where for simplicity I have taken the time increment to be 1. If w_j denotes the mass of particle j then the kinetic energy of the system at time t is given by K(u) = ∑w_ju_j²/2, where u = (u₁,…,u_n).

Let P(x) denote the potential energy of the system, which depends only on the state x. Then the Lagrangian of the system is defined by L(x,u) = K(u)-P(x), and the action of the system is defined as ∑L(x(t),u(t)), where the sum is taken between two fixed time points, which we may take as 0 and T. The principle of stationary action states that the system will take the path that maximizes or minimizes the action. For concreteness, let us assume that the particle takes the path that minimizes the action. Let us denote by S(x,t) the minimum of the sum of ∑L(x(t),u(t)) taken between times t and T, where the minimum is taken over {u(t)}. Then, for each t<T, S(x,t) = min{L(x,u)+S(x+u,t+1)}. This gives us a recursive expression for the function S.

The momentum of particle j is given by p_j = w_ju_j, and the Hamiltonian of the system is defined by H(x,u,p) = pu-L(x,u), where p =(p₁, … ,p_m). Note that from our definition of the Lagrangian above, H(x,u,p) = K(u)+P(x), the energy of the system. We want to derive an expression which relates the functions H and S. From the above we have that S(x,t)-S(x,t+1) = min{L(x,u)+S(x+u,t+1)-S(x,t+1)}. We may re-write this formula using discrete derivatives as -S_t(x,t) = min{L(x,u)+S_x(x,t+1)u}, or alternatively, using the fact that min{a,b}=-max{-a,-b}, as S_t(x,t) = max{-L(x,u)-S_x(x,t+1)u}. This can be written using our definition of the Hamiltonian as S_t(x,t) = max{H(x,u,-S_x(x,t+1))}.We have just derived a discrete version of the so-called Hamilton-Jacobi-Bellman equation.

The continuous time version of this equation is a result of the theory of dynamic programming which was pioneered in the 1950s by the American mathematician Richard Bellman and co-workers. It is very similar in form to the Hamilton-Jacobi equation in classical physics, hence the name. The Hamilton-Jacobi equation was in turn named after the Irish mathematician William Hamilton and the German mathematician Carl Jacobi, and is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton’s laws of motion. The Hamilton-Jacobi equation is considered the ‘closest approach’ of classical mechanics to quantum mechanics, as the Schrödinger equation of quantum mechanics can be derived from it.

The principle of stationary action can also be used in macroeconomic modelling. In this context, we have an economy which produces m commodities, and the state x_j(t) denotes quantity of commodity j produced by the economy t. We also assume that the economy uses n different types of labour, and u_k(t) denotes the quantity of labour of type k employed at time t. Note that unlike in the physical case described above, in general m and n will be different. Furthermore, the ‘mass’ w_k denotes the average wage for labour of type k, and the ‘kinetic energy’ is just the total wage bill: K(u) = ∑w_ku_k = wu. Thus, unlike in the physical case, the kinetic energy is linear in u. The ‘potential energy’ is negative and can be interpreted as depreciation: P(x) =-dx, where d is the depreciation rate.

So the Lagrangian for our economy is given by: L(u) = K(u)-P(x) = wu+dx, which is just wages plus depreciation. Thus the Lagrangian represents the total costs incurred by capitalists. In the physical case, the state of the system evolves according to x(t+1) = x(t)+u(t). The economic case generalizes this to x(t+1) = x(t)+f(x(t),u(t)) for some function f. Just as in the physical case, we assume that production follows the path which minimizes the action ∑L(x(t),u(t)), which is to say that capitalists act in such a way so as to minimize their costs. The ‘momentum’ of commodity j, p_j, is just the price of the commodity, and the Hamiltonian of the system is then defined by H(x,u,p) = pf(x,u)-L(x,u) = pf(x,u)-wu-dx, which represents gross profits, as pf(x,u) = p(x(t+1)-x(t)) represents nominal net output, and wu+dx represents costs.

Clearly, there is more to be investigated here in terms of connections between the two models, but I will leave that for a future blog post.