# Systems of Differential Equations

Business I · Mathematics II · https://tryals.app/learn/business-i/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.

| Eigenvalues | Equilibrium type | Behaviour |
| --- | --- | --- |
| Both real, negative | **Stable node** | Converges to equilibrium |
| Both real, positive | **Unstable node** | Diverges from equilibrium |
| Opposite signs | **Saddle point** | Converges along one direction |
| Complex, negative real | **Stable spiral** | Oscillating convergence |
| Complex, positive real | **Unstable spiral** | Oscillating divergence |

**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.

## Practice questions

6 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. In the Ramsey growth model, the steady state is a saddle point. Why is this economically important?

A. It means the economy always reaches equilibrium regardless of initial conditions
B. It means eigenvalues are complex
C. Exactly one trajectory reaches equilibrium: the saddle path
D. It means the economy is always unstable

**Answer:** C. Exactly one trajectory reaches equilibrium: the saddle path

**Why:** The saddle-path property is the foundation of determinate equilibrium in rational-expectations models. Forward-looking agents pick the unique convergent path, otherwise the economy explodes.

Page: https://tryals.app/practice/business-i/systems-of-differential-equations/in-the-ramsey-growth-model-the-steady-state-is-a-saddle-point-why

### 2. Which spreadsheet formula iterates a linear system: if $x_t$ is in A2 and $y_t$ in B2, with $\dot{x} = 0.3x - 0.1y$ and step $\Delta t = 0.1$?

A. =A2+(0.3*A2-0.1*B2)*0.1
B. =A2+(0.3-0.1*B2)*0.1
C. =A2+(0.3*A2+0.1*B2)*0.1
D. =(0.3*A2-0.1*B2)*0.1

**Answer:** A. =A2+(0.3*A2-0.1*B2)*0.1

**Why:** Euler method: $x_{t+1} = x_t + f(x_t, y_t) \cdot \Delta t$. The formula adds the rate of change times the time step to the current value.

Page: https://tryals.app/practice/business-i/systems-of-differential-equations/which-spreadsheet-formula-iterates-a-linear-system-if-xt-is-in-a2

### 3. Which systems have a stable equilibrium?

A. Both eigenvalues positive (unstable node)
B. Both eigenvalues negative (stable node)
C. Complex eigenvalues with negative real part (stable spiral)
D. One positive, one negative eigenvalue (saddle)

**Answer:** B. Both eigenvalues negative (stable node); C. Complex eigenvalues with negative real part (stable spiral)

**Why:** Stability requires ALL eigenvalue real parts to be negative. Saddle points and unstable nodes have at least one positive eigenvalue.

Page: https://tryals.app/practice/business-i/systems-of-differential-equations/which-systems-have-a-stable-equilibrium

### 4. In a phase diagram, nullclines are curves where one of the two derivatives equals zero.

**Answer:** True

**Why:** The $\dot{x} = 0$ nullcline shows where $x$ is momentarily stationary; the $\dot{y} = 0$ nullcline shows where $y$ is stationary. Their intersection is the equilibrium.

Page: https://tryals.app/practice/business-i/systems-of-differential-equations/in-a-phase-diagram-nullclines-are-curves-where-one-of-the-two

### 5. Match each phase-diagram concept to its description.

**Answer:**

- Nullcline → Curve where one variable is not changing
- Stable manifold → Trajectories converging to a saddle equilibrium
- Phase portrait → All trajectories in the state-space plane
- Saddle path → The unique trajectory rational agents choose

**Why:** Nullclines, manifolds, portraits, and saddle paths, the geometric language of dynamic systems.

Page: https://tryals.app/practice/business-i/systems-of-differential-equations/match-each-phase-diagram-concept-to-its-description

### 6. The Ramsey model's steady state is a saddle point, which sounds like instability. Why do economists treat that as the desirable case?

A. Because exactly one path leads to the steady state
B. Because the model would have no solution if the steady state were stable
C. Because saddle points are more robust to shocks than stable nodes
D. Because instability drives the economy toward higher growth

**Answer:** A. Because exactly one path leads to the steady state

**Why:** The agent chooses initial consumption freely and every choice but one leads to ruin, so saddle-path stability is what makes the optimum unique rather than a menu. It is the standard structure of forward-looking macro models, one predetermined variable, one jump variable, one stable arm.

Page: https://tryals.app/practice/business-i/systems-of-differential-equations/the-ramsey-models-steady-state-is-a-saddle-point-which-sounds-like