FIELD TRANSLATION
Generated equations
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Generated Field Equations

Translation of the microeconomic model into a field representation

Model

Name: Interconnected Investment Model

Type: Dynamic optimization

Original Micro Model

Kt+1=(1−δ)Kt+ItK_{t+1} = (1-\delta)K_t + I_t
It=αKtI_t = \alpha K_t

Micro → Field Mapping

Micro VariableField RepresentationInterpretation
kᵢρ(k,t)distribution of capital across agents
IᵢF(k)drift term

Generated Field Equation

∂ρ(k,t)∂t+∂∂k(ρF(k))=D∂2ρ∂k2\frac{\partial \rho(k,t)}{\partial t} + \frac{\partial}{\partial k} (\rho F(k)) = D \frac{\partial^2 \rho}{\partial k^2}

Equation Components

Density

rho(k,t)\\rho(k,t)

Distribution of agents across capital states.

Drift

F(k)F(k)

Aggregate investment dynamics.

Diffusion

Dfracpartial2rhopartialk2D \\frac{\\partial^2 \\rho}{\\partial k^2}

Uncertainty and heterogeneous shocks.