Question

Write out the characteristic equation for each of the following, and find the character. istic roots: (a) \(y_{t+2}-y_{t+1}+\frac{1}{2} y_{t}=2\) (b) \(y_{1+2}-4 y_{t+1}+4 y_{t}=7\) (c) \(y_{1+2}+\frac{1}{2} y_{t-1}-\frac{1}{2} y_{t}=5\) \(\left(d^{\prime}\right) y_{t+2}-2 y_{t+1}+3 y_{t}=4\)

Short answer

(a) Roots: \( \frac{1 \pm i}{2} \), (b) Root: \( 2 \) (double), (c) Root: \( 1 \), (d) Roots: \( 1 \pm i\sqrt{2} \).

Step-by-step solution

Identify the Homogeneous Part

Start by setting each given equation to zero: (a) \( y_{t+2} - y_{t+1} + \frac{1}{2} y_t = 0 \)(b) \( y_{t+2} - 4y_{t+1} + 4y_t = 0 \)(c) \( y_{t+2} + \frac{1}{2}y_{t-1} - \frac{1}{2}y_t = 0 \)(d) \( y_{t+2} - 2y_{t+1} + 3y_t = 0 \)

Write the Characteristic Equation

Use \( y_t = r^t \) and substitute into each homogeneous equation:(a) \( r^{t+2} - r^{t+1} + \frac{1}{2}r^t = 0 \) simplifies to \( r^2 - r + \frac{1}{2} = 0 \).(b) \( r^{t+2} - 4r^{t+1} + 4r^t = 0 \) simplifies to \( r^2 - 4r + 4 = 0 \).(c) \( r^{t+2} + \frac{1}{2}r^{t-1} - \frac{1}{2}r^t = 0 \) simplifies to \( r^3 - \frac{1}{2}r + \frac{1}{2} = 0 \).(d) \( r^{t+2} - 2r^{t+1} + 3r^t = 0 \) simplifies to \( r^2 - 2r + 3 = 0 \).

Solve for the Characteristic Roots

Solve each characteristic equation for \( r \):(a) \( r^2 - r + \frac{1}{2} = 0 \) using the quadratic formula, \( r = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), yields \( r = \frac{1 \pm i}{2} \).(b) \( r^2 - 4r + 4 = 0 \) can be factored to \((r-2)^2 = 0\), giving \( r = 2 \) with multiplicity 2.(c) \( r^3 + \frac{1}{2}r - \frac{1}{2} = 0 \) is a cubic equation. By factoring or using the rational root theorem, one possible root is \( r = 1 \).(d) \((r^2 - 2r + 3 = 0)\) yields complex roots by solving \( r = \frac{2 \pm \sqrt{-8}}{2} = 1 \pm i\sqrt{2} \).

Interpret the Solutions

Each characteristic root determines the solution type:(a) Roots \( r = \frac{1 \pm i}{2} \) represent a damped oscillation.(b) Repeated root \( r = 2 \) suggests exponential growth or decay.(c) Real root \( r = 1 \) suggests a steady state or unchanging solution.(d) Imaginary roots \( r = 1 \pm i\sqrt{2} \) indicate a growing oscillation.

Key concepts

Difference Equations

Difference equations are mathematical expressions that relate a sequence of values by specific relationships across different time periods or index steps. They are quite similar to differential equations but focus on discrete rather than continuous functions. In simple terms, they help in understanding how a quantity changes over successive time steps. For instance, if you have values of a sequence at point, say \( t \), then the difference equation predicts values for \( t+1 \), \( t+2 \), and so on.

• Used in various fields such as economics, biology, and engineering.
• Assume values only at specific discrete points in time.
• Typically appear in the form, \( y_{t+2} - ay_{t+1} + by_t = c \).

Understanding the setup and solutions of difference equations helps uncover patterns and forecast behavior of dynamic systems.

Characteristic Roots

In the analysis of difference equations, finding the characteristic roots is a crucial step. These roots come from the characteristic equation, which is obtained by substituting the assumed form of the solution, usually \( y_t = r^t \), into the difference equation. The roots reveal the nature of the dynamics involved:

• Solutions for the roots typically give us insights like stability, periodicity, and growth rate of the system.
• An equation like \( r^2 - 4r + 4 = 0 \) provides the roots that are solutions to the homogeneous part of the difference equation.

Solving these roots is typically done using methods like factoring, the quadratic formula, or numerical techniques for more complex, higher-order polynomials. Each set of roots has a corresponding general solution which informs how the sequence behaves over time.

Complex Roots

Complex roots often appear when dealing with the characteristic equations of difference equations. When you solve a characteristic equation and find roots that have imaginary components, they are called complex roots. Complex roots typically arise from the discriminant of the quadratic equation being negative, which suggests oscillatory behavior:

• Expressed usually in the form \( r = a \pm bi \), where \( i \) is the imaginary unit.
• Lead to solutions of the form \( C \cdot e^{at}\cdot(\cos(bt) + i\sin(bt)) \) indicating periodic or oscillating solutions.
• These solutions often represent systems with cycles or repetitive patterns.

Handling complex roots is fundamental for understanding dynamic systems involving cyclical processes, like electrical circuits or certain economic models.

Exponential Growth

Exponential growth is a type of behavior shown by certain solutions of difference equations where the quantity increases (or decreases) at rates proportional to their current value. In relation to characteristic roots, a repeated real root, such as \( r = 2 \) from \((r-2)^2 = 0\), suggests exponential behavior:

• Describes scenarios where processes multiply rather than add, common in populations or viral spread studies.
• Solutions showing exponential growth take the form \( C \cdot r^t \), where \( C \) is a constant based on initial conditions, and \( r \) is a characteristic root.
• Exponential growth may also appear as decay if the root is a fraction less than one.

Recognizing exponential growth helps in predicting future states for economies, populations, or even investments, essential for strategic planning across industries.