Comments on the generalized RCK model

More final, final notes on the Ramsey-Cass-Koopmans model

This post (repeats, then) continues on the technical discussion of the generating function.

We had derived a generating function G(k,pₖ,s). First requiring that k˙=f(k)-c, and recognizing c=u⁻¹(s), the most general form of G satisfying our dynamic conditions is

\(\frac{\partial G}{\partial p_k}=\dot{k} = f(k)-u^{-1}(s) \implies G=p_k(f(k)-u^{-1}(s)) + g(k,s)\)

Plugging this into the dynamic relation for pₖ gave us

\(\dot{p_k} = -\frac{\partial G}{\partial k} - p_k\frac{\partial G}{\partial s} =-p_kf'(k)-\frac{\partial g}{\partial k} - p_k\left(\frac{\partial g}{\partial s}-p_k\frac{\partial }{\partial s}u^{-1}\right)\)

We made the ansatz: in the limit the last term in brackets is constant so p*ₖ∝ u’, so:

\(\frac{u''}{u'}\frac{dc}{dt}=\frac{\dot{p^*_k}}{p^*_k} =-f'(k) + \left(\rho_1 - \frac{\partial g}{\partial s}\right) -\frac{1}{p^*_k}\frac{\partial g}{\partial k} \qquad (KR^*)\)

so that when we chose g such that ∂g/∂k=0 and ∂g/∂s=ρ₂ (constant) and set ρ=ρ₁-ρ₂, we recovered the Keynes-Ramsey condition (KR), and saw G=pₖ[f(k)-u⁻¹(s)] + ρ₂(s-s*).

Observations

Here I want to say that those last choices for g only need to hold in the limit to recover (KR). For example, g could depend on (k-k*)² and then ∂g/∂k=0 when k=k* but not in general! (We can find k* in terms of s* by solving the (KR) condition). Similarly, g could also depend on higher order terms of (s-s*)—so could even be given by e.g. exp(ρ₂(s-s*)).

This could lead to much more interesting dynamics for G, since

\(\frac{dG}{dt}=-\frac{\partial G}{\partial s}G = -\left( \frac{\partial g}{\partial s}-\frac{p_k}{u'} \right)G\)

and this is really only easy to solve if the coefficient (between brackets) is constant. The limit is still G=0, but the way it gets there could be chaotic¹—literally. Similarly, s˙=-g(k,s) can be anything, though at the limit s˙=-const · s so that s is exponential.

Comparisons

Superficially, G=pₖk˙+g(k,s) looks like the Hamiltonian used to solve the RCK, H=pₖk˙+L(k,k˙,t):=pₖk˙+e⁻ᵖᵗu(c)². But the role is quite different: the Lagrangian depends on flows k˙ (and possibly time t) the undetermined function g only on ‘utility’ s. I interpret the Lagrangrian as a cost function to be integrated along a path—and the system assumed to choose the path of lowest cost—whereas g determines how the system discovers and approaches that optimal path.

Extensions

The reasoning extends straightforwardly to multiple dimensions. The RCK has only one state variable k, and one associated co-state variable pₖ=μ. But we could ‘endogenize’ labor by including variables l and pₗ in G(k,l,pₖ,pₗ,s) and requiring:

\(\dot{l} = \frac{\partial G}{\partial p_l}, \quad \dot{p_l} = -\frac{\partial G}{\partial l} -p_l \frac{\partial G}{\partial s}, \quad \dot{s} = p_k\frac{\partial G}{\partial p_k}+p_l\frac{\partial G}{\partial p_l}-G\)

and so forth for any other variable we need to include to properly model states. That we can formulate the model is only a first step though; solving such a system (now 5 coupled partial differential equations) gets complicated rather quickly³.

1. Even more so if production f(k) and utility u(c) are not time-invariant but depend on t as well, through changing technology or tastes.

2. Earlier, I had followed the economic convention of writing it in this form and requiring as one of the first-order conditions that ∂H/∂c=0. We could also subsitute c=f(k)-k˙ and require

\(\frac{\partial H}{\partial \dot{k}}=0\implies p_k =\frac{\partial L}{\partial \dot{k}}=\pm e^{-\rho t}u'\)

(give or take a minus sign) which specifies the co-state variable pₖ=μ.

3. Unless the system permits the separation of variables (Wiki).