Consider an economy which produces m commodities using n types of labour. For such an economy we can define a commodity vector as a vector with positive elements of length m, and a labour vector as a vector with positive elements of length n. The economy is defined by an activity set, with the interpretation that an element (x,u,x’) of this set represents a possible configuration of commodity inputs x, labour inputs u, and commodity outputs x’ for the economy. For this economy, a price vector is a vector with positive elements of length m, and a wage vector is a vector with positive elements of length n. The gross profit associated with activity (x,u,x’), price vector p, and wage vector w, is given by the equation: H = p(x’-x)-wu; and the associated profit rate is given by: r = H/px.
Rearranging the above equations gives: px’ = px(1+r)+wu. The production price vector associated with the activity (x,u,x’), wage vector w, and profit rate vector r, is defined as the price vector p satisfying this equation. Rearranging this equation again gives: wu = px’-px(1+r). The wage-price curve associated with the activity (x,u,x’) and price vector p is the function w() which satisfies this equation, considered as a function of the profit rate: w(r)u = px’-px(1+r). This demonstrates an inverse relationship between wages and profit rates. Note also that our expression for gross profits, H = p(x’-x)-wu, bears a striking resemblance to the expression for the Hamiltonian from physics: H = pdx/dt-L. Here, L is the Lagrangian, defined as the difference between a system’s kinetic and potential energy.
In physics, the action, S, of a physical system is defined as the integral of the Lagrangian L over a given time interval: S = ∫ Ldt. The principle of stationary action, also known as Hamilton’s principle, states that the physical path taken by a system between two points in time is the one for which the action integral is stationary (either a minimum, maximum, or saddle point). This principle determines the dynamics of the system. If x denotes the state of the system then the conjugate momentum is defined as the partial derivative of L with respect to dx/dt. It can then be shown that the dynamics of the system follows what are known as Hamilton’s equations: dp/dt = -∂H/∂x and dx/dt = ∂H/∂p, where H = pdx/dt-L is the Hamiltonian. This is a reformulation of Newtonian mechanics.
In engineering, the Hamiltonian describes not the dynamics of a system but conditions for minimizing some scalar function thereof (the Lagrangian) over time with respect to a control variable, u. In this context, the dynamics of the system are given by dx/dt = f(x,u) for some function f(), and the aim is to minimize the action functional S = ∫ Ldt over all control trajectories {u(t)}. The Hamilton is then given by: H = pf(x,u)-L. According to what is known as Pontryagin’s maximum principle, the optimal control trajectory (that is, the control trajectory that minimizes S) must also maximize the Hamiltonian H at all time points t. Furthermore, necessary conditions for the Hamiltonian to be maximized are given by the equations: dp/dt = -∂H/∂x, dx/dt = ∂H/∂p, and ∂H/∂u = 0. The first two are identical to Hamilton’s equations in physics.
Returning to the economic context set out above, we may identify the control variable u with the labour input; the Lagrangian L with the associated labour cost: L = wu; and the Hamiltonian H with our expression for gross profit: H = p(x’-x)-wu. (Note that we are now working in a discrete rather than a continuous time framework.) We may also define the labour value of the commodity vector y as the minimum value of L = wu subject to the constraint that there exist commodity vectors x and x’ such that (x,u,x’) is an element of the activity set, x’-x is greater than or equal to y, and (x,u,x’) maximizes gross profits. Then, considering Pontryagin’s maximum principle, if {u(t)} is the labour input trajectory that attains the minimum value for each t subject to these constraints, we would expect this trajectory to minimize the functional S = ∑ L; that is, to minimize the sum of all labour values over a given time period.
This suggests that the striking resemblance between our expression for gross profits and the expression for the Hamiltonian from physics is no mere coincidence. Just as the time evolution of a physical system is governed by the principle of stationary action, our analysis above suggests that the time evolution of an economic system is governed by what might be called the ‘principle of minimum labour values’. By Pontryagin’s maximum principle, a necessary condition for the minimum to be achieved is that profits are maximized at each time step. This should not be a surprise as one of the key characteristics of capitalism – perhaps the defining characteristic – is that firms always aim to maximize profits. The principle of minimum labour values might help shed some light on why this is so.