Systems of Differential Equations

When Variables Interact

Many economic systems involve two or more variables that evolve simultaneously and influence each other. A system of ODEs captures this:

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

Linear systems: when $f$ and $g$ are linear, the system can be written in matrix form:

$$\dot{\mathbf{x}} = A\mathbf{x}, \qquad \text{where } A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

The solution involves the eigenvalues of $A$, connecting this lesson to Lesson 5:

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

Phase diagrams show trajectories in the $(x, y)$ plane — how the system evolves from any starting point. The equilibrium $(x^*, y^*)$ is where $\dot{x} = 0$ and $\dot{y} = 0$ simultaneously.

Classification by eigenvalues:

• Both $\lambda < 0$: stable node — all trajectories converge to equilibrium.
• Both $\lambda > 0$: unstable node — trajectories diverge from equilibrium.
• $\lambda_1 < 0 < \lambda_2$: saddle point — stable in one direction, unstable in another.
• Complex eigenvalues with negative real part: stable spiral — oscillating convergence.
• Complex with positive real part: unstable spiral — oscillating divergence.

Economic examples:

• IS-LM dynamics: output and interest rate adjust simultaneously — output responds to interest rates, interest rates respond to money demand (which depends on output).
• Predator-prey in markets: firms and consumers interact — market share dynamics where one firm's gain is another's loss.
• Ramsey growth model: consumption and capital evolve jointly — the saddle-path solution is the unique trajectory leading to the steady state.

The saddle path is especially important in economics: forward-looking agents (who expect the economy to reach equilibrium) choose the initial conditions that place the system on the stable manifold of the saddle point. This is the mathematical foundation of rational-expectations macroeconomics.

Eigenvalues decide the fate

Tip: You rarely need the full solution — the eigenvalues alone classify the equilibrium. Real parts answer "stable or not"; imaginary parts answer "does it oscillate." That is usually the whole economic question.