🌐 Econ-Sim

Introduction

This simulator implements a networked intertemporal optimization model — a Ramsey-style social planner's problem generalized to a graph-structured economy. The model combines heterogeneous agents (households with different preferences) and network geography (goods move along graph edges) into a unified framework with deep analogies to lattice field theory in physics.

The Mathematical Model

The Planner's Objective

The planner maximizes welfare over an infinite horizon (approximated by finite T):

Where β ∈ (0,1) is the discount factor, ωᵈ are welfare weights (Pareto weights tracing the efficiency frontier), and uᵈ(c) is household d's utility from consumption vector c at time t.

Constraints

Node-level flow balance (conservation on the graph):

Production + net transfers in - consumption - change in stock = 0, at every node, every period. This is the economic analogue of a continuity equation on a graph.

Network stability condition:

The spectral radius of the adjacency matrix must be strictly below one. This ensures the Neumann series (I - A)⁻¹ = Σₖ Aᵏ converges — the network's "Green's function" exists.

First-Order Conditions

Attaching multipliers λₜ⁽ⁱ⁾ (node balance) and μₜ⁽ᵈ⁾ (goods accounting):

Consumption:

Welfare-weighted marginal utility equals shadow price.

Transfers:

Shadow values equalize across connected nodes — a no-arbitrage condition in space.

Euler Equation (Ramsey condition):

The shadow value of resources today equals their discounted shadow value tomorrow, gross of the marginal product of storage. This is the intertemporal no-arbitrage condition.

The Physics Analogy

To a physicist, the economic Lagrangian is a familiar object in unfamiliar dress. It has exactly the structure of a classical field theory — but defined on a discrete spacetime and extremized for a different reason.

Physics (Continuum)	Economics (These Notes)
Field φ(x,t)	Consumption c, transfers T, stocks S at node-time (i,t)
Spacetime point (x,t)	Node-date pair (i,t) — a lattice, not a continuum
Action S[φ] = ∫dt dx ℒ	Welfare W = Σₜ βᵗ{...}
Lagrangian density ℒ	Per-period welfare plus constraint terms
Time integral ∫dt	Discrete sum Σₜ₌₀^∞
Space integral ∫dx	Sum Σᵢ over nodes (space is a graph)
Multiplier field λ(x,t)	Multiplier "fields" λₜ⁽ⁱ⁾, μₜ⁽ᵈ⁾ on the node-time lattice
Euler-Lagrange equation	Discrete Euler equation λₜ = β(1+MP)λₜ₊₁
Conjugate momentum p = ∂ℒ/∂q̇	Shadow price λ = ∂V/∂S (envelope theorem)
Hamilton-Jacobi equation	Bellman equation V(S) = max{u(c) + βV(S')}
Green's function (I-A)⁻¹ = Σₖ Aᵏ	Network resolvent as path expansion

Discounting as Euclidean Tilt

Physical actions are time-translation invariant, and Noether's theorem delivers conserved energy. The economist's factor βᵗ is an explicit time dependence: the future literally counts less. The discrete Noether charge fails to be conserved; in its place the Euler equation prescribes a drift of the shadow price.

Taking the continuous-time limit with discount rate ρ = -ln(β):

This is mathematically the same as Wick-rotating to Euclidean time with an imaginary-time-dependent potential. The model lives in Euclidean signature — relaxation, not oscillation.

First-Order Relaxation vs Second-Order Waves

Physical Lagrangians contain kinetic terms φ̇², producing second-order Euler-Lagrange equations: waves, oscillation, propagation. The economic Lagrangian contains ΔS linearly, inside a constraint. The resulting dynamics are first-order difference equations — relaxation, not ringing. Economic systems decay toward their optima like solutions of a diffusion equation.

Description vs Prescription

One genuine difference remains. In Hamilton's principle, nature extremizes the action and the Euler-Lagrange equations describe what happens. In Ramsey's problem, the planner chooses the path that maximizes welfare, and the Euler equation is a normative optimality condition — a prescription, not a law of motion. The calculus-of-variations machinery is identical; its ontological status is reversed.

Interactive Examples

Try these pre-configured scenarios in the simulator:

• Star Network: One central producer feeding multiple households. Observe how shadow prices equalize across connected nodes.
• Chain Network: Linear chain of producers. Watch how consumption propagates and dissipates.
• Two Islands: Disconnected subgraphs. Note different welfare levels when networks cannot trade.