A Special Theory of Employment, Interest, and Money

Abstract

We study a supply-side macro-financial dynamical system in which output velocity V=Y is an independent state variable governed by ˙V = (A/∆)V, where A(·) is effective productive capacity and ∆ = λ(Y−Y₀)−A(·) is an endogenous threshold gap. Output thereby carries momentum, and the threshold surface Σ : {∆ = 0} separates self-amplifying from self-dampening regimes. Exact results (frozen-A∗ reduced fast subsystem). The apparent singularity at ∆ = 0 is removable by the time reparametrisation dt/dθ= ∆, which yields a smooth desingularised vector field with closed-form solutions for V(θ), y(θ), and physical time via the exponential integral Ei. This delivers a complete classification of four trajectory types and a proof that V is sign-invariant: within this subsystem, oscillations cannot originate in the real sector and must enter through finance. On the SC−branch, Σ is not merely a removable parametrisation artefact but a genuine finite-time blow-up boundary (V →−∞ in finite physical time). Local linear and Hopf results (reduced financial block˜M). The linearisation J∗ is block- lower-triangular with a non-semisimple double-zero eigenvalue in the (Y,V,L) block; transversal stability requires˜M Hurwitz, established via the centre-manifold theorem. For the 3×3 financial sub-block ˜M governing (Db,E,Bk), we derive explicit Routh–Hurwitz conditions and a Hopf criterion in closed form. The first Lyapunov coefficient l1 admits an Ω-factorisation under βK = 0, reducing the subcritical/supercritical question to a phase-angle condition. Exact comparative statics, demand closure, and impulse theory. The equilibrium Jacobian for the financial triple equals˜M exactly, so the implicit function theorem yields closed-form total derivatives for all four policy instruments; signed results are unconditional when˜M is Hurwitz. The system is closed on the demand side with payroll taxes, unemployment benefits, household wealth, and an aggregate demand constraint; exact results include sign-switching of V, demand-consistency of the SC−spiral near Σ, benefit-floor stability via a demand-constrained Jacobian, and impulse responses showing that both supply-side and discount-rate shocks ring the financial block at ω₀, discriminable only by their harmonic fingerprint. A proposed OGY–MPC stabilisation architecture (a design, not a proved global theorem) completes the policy framework. Throughout, every result is stated with its scope of validity: exact theorems for the reduced subsystem, conditional results for the full 14-dimensional model.