A generating function for the RCK model

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

This post re-ups the series on economic geometry; since there is a math student helping me out. Here’s where I left off last time.

The goal of this post is to find a generating function G(k,pₖ=μ,s)¹ for the Ramsey-Cass-Koopmans model (Wiki), so that the dynamics given by the generating function²

\(\dot{k} = \frac{\partial G}{\partial p_k}, \quad \dot{p_k} = -\frac{\partial G}{\partial k} -p_k \frac{\partial G}{\partial s}, \quad \dot{s} = p_k\frac{\partial G}{\partial p_k}-G \qquad (*)\)

correspond to the RCK model, and that G=0 implies the Keynes-Ramsey rule (Wiki):

\(f'(k) + \frac{u''}{u'} \frac{dc}{dt} = \rho \qquad \qquad (KR)\)

So, let’s get to it!

The first thing to remark is that the RCK model specifies k˙=f(k)-c, the change in capital is the current production minus what is consumed. To describe c in terms of our variables, remember in RCK consumption c provides utility u(c), and our s encodes that utility—so we can impose s=u(c) or equivalently c=u⁻¹(s)³. Then using (*)

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

in general, where g(k,s) could be any function that does not depend on pₖ.

Now we will consider this in the second dynamic relation in (*), and derive

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

Ugly. But. We see f’(k) appearing, which is hopefull. We also know ∂u⁻¹/∂s = 1/u’⁴.

Now let me forego generality and make an ansatz. That is that in the optimal (limit) situation, the last term in brackets is constant. That is

\(\begin{align*} p^*_k\frac{\partial u^{-1}}{\partial s} = \frac{p^*_k}{u'}=\rho_1 \quad &\implies p^*_k = \rho_1 u', \\ \text{and so} \quad &\implies \dot{p^*_k} = \rho_1 u'' \cdot \frac{dc}{dt} %\dot{c} \end{align*}\)

(where I have reintroduced c so you can guess where this is going). Specifying our dynamic relation for pₖ to the optimal path:

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

so then if we were to choose g such that ∂g/∂k=0 and ∂g/∂s=ρ₂ (constant) and define ρ=ρ₁-ρ₂, we recover the (KR) condition above! Then G=pₖ[f(k)-u⁻¹(s)] + ρ₂(s-s*). and finally,

\(\dot{s} = -g(k,s) = \rho_2(s^*-s) \implies s(t) = s^* \left(1- e^{-\rho_2 t}\right)\)

if we suppose s(0)=0. The system captures “free” utility approaching its limit s* at rate ρ₂. The societal discount rate ρ of (KR) is the time preference ρ₁ corrected for this rate of technology growth ρ₂. G=0 and s=s* imply c*=f(k*), all additional production is consumed.

So, where does this bring us?

Why all this technique to reproduce something we already knew?

• Most importantly, we have generalized the model to non-optimality. By considering different functions g(k,s) we can specify how the system approaches its optimal path.

• We have—in the illustrated case—distinguished productivity gains ρ₂ from pure time preferences ρ₁, and deduced that the societal discount rate satisfies ρ=ρ₁-ρ₂.⁵

• We have not required (inter-temportal) utility maximization! The dynamics of the generating function then ensure that “free” utility is captured until G=0.

1. Notation, notation: I will use pₖ for the utility-price of capital k; the economics literature often uses μ and calls it the co-state variable.

2. Derived here, if you’re so inclined.

3. Fortunately, a utility function is invertible as it is monotonic (u’>0) and convex (u’’<0).

4. Differentiating both sides of u(u⁻¹(s))=s with respect to s, we see u’(…)·(u⁻¹)’=1

5. I am not confident enough in my macroeconomics to write this as r-g.