Systems of Differential Equations

Interacting Variables

Many economic systems involve variables evolving simultaneously. A system of ODEs captures this interaction:

$$\dot{x} = f(x, y), \qquad \dot{y} = g(x, y)$$

When linear, we write it in matrix form as $\dot{\mathbf{x}} = A\mathbf{x}$, where $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

The general solution relies on the eigenvalues of $A$:

$$\mathbf{x}(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2$$

Phase diagrams plot trajectories in the $(x, y)$ plane. The equilibrium $(x^*, y^*)$ occurs where $\dot{x} = 0$ and $\dot{y} = 0$ simultaneously.

Economic examples include IS-LM dynamics, predator-prey market share models, and the Ramsey growth model where consumption and capital evolve jointly.

Eigenvalues and Stability

You rarely need the full solution because eigenvalues alone classify the equilibrium. Real parts answer if it is stable; imaginary parts answer if it oscillates.

The saddle path is crucial in macroeconomics: forward-looking agents select initial conditions placing the system on the stable manifold, driving rational-expectations models to steady state.