Exercises 17 and 18 concern a simple model of the national...

Exercises 17 and 18 concern a simple model of the national economy described by the difference equation $$ Y_{k+2}-a(1+b) Y_{k+1}+a b Y_{k}=1 $$ Here $Y_{k}$ is the total national income during year $k, a$ is a constant less than $1,$ called the marginal propensity to consume, and $b$ is a positive constant of adjustment that describes how changes in consumer spending affect the annual rate of private investment. Find the general solution of equation $(14)$ when $a=.9$ and $b=\frac{4}{9} .$ What happens to $Y_{k}$ as $k$ increases? [Hint: First find a particular solution of the form $Y_{k}=T,$ where $T$ is a constant called the equilibrium level of national income. 1

Exercises 17 and 18 concern a simple model of the national economy described by the difference equation
$$
Y_{k+2}-a(1+b) Y_{k+1}+a b Y_{k}=1
$$
Here $Y_{k}$ is the total national income during year $k, a$ is a constant less than $1,$ called the marginal propensity to consume, and $b$ is a positive constant of adjustment that describes how changes in consumer spending affect the annual rate of private investment.

Find the general solution of equation $(14)$ when $a=.9$ and $b=\frac{4}{9} .$ What happens to $Y_{k}$ as $k$ increases? [Hint: First find a particular solution of the form $Y_{k}=T,$ where $T$ is a constant called the equilibrium level of national income. 1

Key Concepts

Characteristic Equation

The characteristic equation is obtained by assuming a solution of the form r^k for the homogeneous part of the difference equation. This transforms the recurrence relation into a polynomial equation, where the roots determine the nature and behavior of the solutions, such as their oscillatory properties and growth or decay rates.

Equilibrium Level

The equilibrium level in a difference equation is a constant solution where the system does not change over time. In economic models, this represents a steady state of national income where inputs balance outputs, and the system remains stable unless disturbed by external factors.

Stability Analysis

Stability analysis involves examining the roots of the characteristic equation to determine the long-term behavior of the system. If all roots lie within the unit circle (i.e., their absolute values are less than one), then the system is stable and solutions converge to the equilibrium level. If any root lies outside the unit circle, the solution will diverge over time.

Difference Equations

Difference equations are mathematical models used to describe sequences where each term is defined in relation to previous terms. They are the discrete counterpart of differential equations and are fundamental in modeling dynamic systems in economics, biology, and engineering.

Particular and Homogeneous Solutions

To solve a linear non-homogeneous difference equation, one typically breaks the solution into two parts: the homogeneous solution, which satisfies the corresponding homogeneous equation, and a particular solution, which is a specific solution to the full non-homogeneous equation. This approach is essential because it allows the construction of the complete general solution as the sum of these components.

Transcript

00:01 In this video, we're dealing with a model, which is a difference equation displayed here, and critically, this is a non -homogeneous difference equation due to the equals 1 on the right -hand side.

00:12 We're going to work with this model specifically when the coefficient of a is 0 .9, and b is 4 .9th, and so the difference equation then becomes y k plus 2, minus a times 1 plus b, in this case, becomes 1 .3, then it's multiplied by, y k plus 1 and a times b becomes 0 .4 so we write plus 0 .4 times y k equals 1 so next our goal here is to find the general solution to the model for these specific constants a is 0 .9 and b is 4 9th and so to that end we need to find a particular solution since the right -hand side is equal to a constant 1 we suppose that c or we can say let c be a constant and assume that this constant c solves the difference equation.

01:14 If we make that assumption and insert c in place of y to the power k plus two or y sub k plus 2 and so on, we obtain the equation that c minus 1 .3 times c plus 0 .4 times c is equal to the constant 1.

01:31 And if we collect all like terms, we have 0 .1 times c.

01:36 Is equal to 1 and this finally implies that the constant c must be equal to 10.

01:43 What this work shows so far is that yk set equal to the constant 10 is a particular solution of this difference equation displayed here.

02:01 So now that we have a particular solution we can obtain all solutions by converting to the homogeneous equation.

02:08 It's y k plus 2 minus 1.

02:11 0 .3 times yk plus 0 .4 times yk, or it should be a k plus 1 plus 1 1 here, equals 0.

02:22 The auxiliary equation for this is going to be r squared minus 1 .3 times r plus 0 .4 equals 0.

02:33 If we plug this auxiliary equation into the quadratic formula, we find then that the solutions are r equals 0 .5, and 0 .8...

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